Introducing into this equation the value of t=0.043, we find λ= 61.05 (19) √ pd Combining equation (19) with equation he gives it the form p=15,547,000 p= τα Assuming that the preceding investigations indicate that the collapsing pressure of tubes is given by an equation of the form p=c tn 10.9 1.16 and, taking the experiments on thin tubes as a starting point of the comparison, Unwin deduces values of the factor c and the exponent n from a number of experiments with thicker tubes, combining the results obtained with tubes of similar construction. From two experiments with tubes made with a longitudinal lap-joint, [marked (a) and (b), in section 2], he deduces the formula p=7,363,000 p= 9170 10.9 1.16 (20) Fairbairn's formula for the same tubes, is 9838 (21) Unwin says, regarding these formula: "It will be seen that the indices of the which is not widely different, though thickness do not differ more than would obtained in an entirely different manner. be anticipated from Hodgkinson's value. Unwin next defines the limits of length It is believed that the separation of the and thickness within which the formula experiments, here adopted, into sets of is applicable. He finds, max. 6.7 d similar experiments leads to more relia12.22 ble results than the mixing up of heteronearly; min. 4468 The two do.18 geneous experiments to obtain average d values of constants. In applying these formula to practical cases, it ought to be 19 the circular form may greatly affect the borne in mind that slight deviations from value of A, and much reduce the strength of the flue." limits coincide when t= For greater values of t the formulæ cease to be applic able. 12.1 10.91.16 p=15,547,000 (24) 12.35 20.91.16 In order to make the latter formula convenient for arithmetical calculations, (22) From one experiment with a tube having a longitudinal butt-joint, [marked (b) in section 2], he gets the formula 12.21 p=9,614,000 0.9 1.16 (23) From five experiments with tubes having longitudinal and circumferential joints [marked (c), (k), (l), (m) and (n), in section 2], he derives the formula where a ẞy are variable co-efficients, the values of which, corresponding to certain values of t, l'and d, are given by him in a table adjoined to his investiga tion. pl S-Pl 2 t Theodore Belpaire has made an attempt to develop a formula for the strength of flues from experimental data, from which the variable and uncertain element of strength due to circularity of form has been eliminated. He starts with the idea that flues derive their main strength from the rigid fastenings of their ends. Considering the flexure of a narrow strip of a tube cut parallel to its axis, he finds that the bending forces are so insignificant that they may be neglected without his investigation on the equation giving sensible error; and he bases, therefore, the greatest shearing forces which exist at the rigidly secured ends of the loaded strip, viz.: (26) The highest value of S consistent with stability is to be deduced from experimental data. Belpaire rejects all experiments with circular flues, and uses those made with elliptical ones, because in the latter cases the relation existing between the strength and the form of the transverse section can be deduced with some degree of certainty. Calling D the greater and X the lesser diameter of the ellipse, he represents its all tubes derive more or less additional eccentricity by strength from these causes, he considers one-quarter a sufficiently large factor of safety in proportioning flues by means of formula (33). (See "Note sur la résistance des tubes pressées de l'extérieur," par Théodore Belpaire, in "Annales du Génie civil," Mars, 1879). D-X (27) S is evidently a function of the eccentric- e= (28) Since the resistance to flexure approaches zero, and consequently. S=0, when E=1, and since, on the other hand, S a maximum, when e=0, the equation (28) can be put under the following form, viz.: S=A (1—e3.) (29) S (and consequently A) is likewise a function of and an approximate value for A may be deduced from an equation t ď of the form A=m+ » (2) + 2()' Since S=0, when =0, 2 S= { "(†) +2 (†) * } (1—e') (31) Belpaire deduces values forn and from two experiments by Fairbairn on elliptical tubes. Introducing the values thus found into equation (31), and making e=0, he gets the following expression for S applicable to tubes of a circular cross-section, viz.: 2 S=1,713,576 (4)—28,446,200 (*)* (32) Introducing this value of S into formula (26), we get p=3,427,152- -56,892,400- (33) Equation (33) may be written as follows, viz: p=3,427,152 ld 1-16.6 ia ( The factor (1 — 16.6 6 becomes a) (33a) determined. 16 D. K. Clark, in his "Manual of Rules," (30) etc,, p. 696, gives the dimensions of six flues, selected from the reports of the Manchester Steam-Users Association. 1862-69, which collapsed while in actual use in boilers. These flues varied from 24 to 60 inches in diameter, and from 3 to inch in thickness. They consisted of rings of plates riveted together, with one or two longitudinal seams, but all of them unfortified by intermediate flanges or strengthening rings. At the collapsing pressures the flues experienced compressions ranging from 1.53 to 2.17 tons, or a mean compression of 1.82 tons per square inch of section. From these data Clark deduced the following formula "for the average resisting force of common boiler flues," viz : p=t (50,000-500) (34) where p is the collapsing pressure in pounds per square inch, and d and t are the diameter and thickness expressed in inches. It is assumed that the flues are not strengthened by rings. The influence of length on the strength of flues Clark calls an uncertain element; it is, however, a very important one, as proven by Fairbairn's experiments, and for this reason the formula is not generally applicable. t consequently p=0, when 16.6=1, or d= 16.6 t. This indicates that the formula is, at best, applicable only within certain limits, which, however, have not been t1 ld t3 ld Applying this formula to a number of examples of tubes collapsed under known conditions, Belpaire finds that the actual collapsing pressures were from 1.05 to 4.15 times as great as the pressures given by formula (33). These variations he ascribes to the influence of the uncertain elements of stiffness, circularity of form and homogeneity of material, which, he avers, should not be allowed to influence The following table contains the data of all experiments on the collapse of a formula of this kind. Since, however, flues having thickness of one-eighth inch and more, of which a reliable record could large results, except in the case of No. be obtained, and which are described in 11, where the agreement is tolerably section 2 of this article; also the col- close. lapsing pressures calculated by means of the different formulae, of which an account has been given. In comparing the calculated with the actual collapsing pressures, for the purpose of estimating the reliability of the several formulæ, it is, of course, necessary in each case not to take into consideration those experiments from which the formula was deduced. In Fairbairn's formula (2) the index number of the power of t was deduced from the results of experiments Nos. 1, 2 and . Leaving these experiments out of consideration, only one experiment, viz., No. 11, shows a tolerably close agreement between the actual and the calculated collapsing pressures; all other experiments give too large a value for flues having a length less than 60 inches, and too small a value for flues having a length of 276 inches or more. Fairbairn's approximate formula (4) gives in every case a larger value for the collapsing pressure than formula (2), as previously explained. It is easily seen that the close agreement between the actual and calculated results in the case of experiments Nos. 9 and 10 is purely accidental. Experiments Nos. 4 and 5 were rejected by Fairbairn and Unwin, but perhaps not on sufficient grounds. Nystrom's formula (11) agrees quite well with the results of these experiments. Belpaire's formula (33) gives in only one case a close agreement with the actual results of the experiments; and the degree of agreement varies widely in the other cases. Grashof's formula No. (9) gives in every case much too high a value of p, except in experiments Nos. 1, 2, 5 and 6, from which the formula was deduced. Nystrom's formula (10) was deduced from Fairbairn's experiments, but it is not stated which of them were used. As in no case the tensile strength of the metal of which the experimental flues were made has been recorded, an average value of T=50,000 in the case of iron flues, and of T=70,000 in the case of the steel flue, has been assumed. On the whole, this formula gives a closer agreement of the calculated with the actual collapsing pressures in experiments on flues of every description than any of the other formulæ. 4. INSUFFICIENCY OF THE ABOVE-DESCRIBED EXPERIMENTS.-It will be noticed that many of the formulæ given in the previous section, though deduced mostly from the same set of experiments, and being similar in form, assign very different values to the influence of length, diameter and thickness of tubes on their resistance to collapse, and that most of them give very discordant results. The differences in the two formulæ, (8) and (9), deduced by Grashof from Fairbairn's experiments, illustrate plainly the fact that the numerous experiments on very thin tubes are of relatively little value for determining the strength of flues used in steam boilers. In fact, all those formulæ which are based principally upon the experiments with tubes 0.043 inch thick, are very defective. This remark applies specially to Fairbairn's formula (2), to Love's formula (11), and also to Unwin's formulæ (22), (23) and (24). Unwin recognized the fact that it was hazardous to apply to thick tubes the rules relating to the influence of length and diameter on the strength deduced from very thin tubes; but he was compelled to do so by the paucity of available experiments. He derived the numerical factor and the index number of t in formula (22) from two experiments, those in formula (23) from one experiment, and those in formula (24) from five experiments, on tubes from inch to inch thick. There are not a sufficient number of experiments extant to test fully the value of his formulæ. Love's formula (11) gives in every case too large a value, probably because it was deduced mainly from Fairbairn's experiments with very thin tubes. It should be especially noticed that Unwin's formulæ (22), (23), (24), ap-not a single experiment has been made plied to other experiments than those on tubes representing exactly the ordifrom which they were deduced, give too nary construction and dimensions of cy TABLE SHOWING DIMENSIONS, AND ACTUAL AND CALCULATED COLLAPSING PRESSURES OF FLUES. I. II. III. IV. V. VI. VII. VIII. IX. X. XI. XII XIII. XIV. XV. XVI. XVII. XVIII. XIX. XX, XXI. Calculated Collapsing Pressure. Number of experiment. ~Length in inches. Pss 331* Fair 378* Fair 359* 258 Fair 0.79 146 0.35 Longitudinal lap-joint, 26 0.24 Longitudinal & transverse joints [n.] 1.18 51 0.49 Long'tud'l & transverse j'nts [f] 1.34 101 0.77 Longitudinal butt-joints [g.] 1.54 128 0.64 Longitudinal butt-joints (welded) [h.] lindrical furnace flues, so extensively for their object the determination of the used at the present day in marine boil- influence which the following elements ers. The two flues, marked (f) and (g), have on the strength of flues, viz.: 1. used in the experiments made at the The length, diameter and thickness, with Washington Navy Yard in 1874 (See the proportions usual in practice. 2. Section 2 and Nos. 11 and 12 in the table), Various modes of construction, e. g., the were made of very thin metal relatively use of riveted butt-joints and of welded to their diameter. The flue used in the joints for the longitudinal seams, differ. experiment made by the Leeds Forge ent styles of circumferential joints, and Company (See Section 2, and No. 13 in different methods of securing strengththe table), while being otherwise an ex- ening rings. 3. The kind and quality act representation of a cylindrical fur- of the metal of which the flues are made, nace flue, was not secured at the end; viz., iron or steel, and especially the ducand there is no doubt that this circum- tility and limit of elasticity of the stance altered materially the conditions metal. 4. Various elements of weakof the experiment, and was one cause of ness common in practice, especially devithe low collapsing pressure of this tube. ations from the circular form, and local The influence of deviations from the weaknesses produced by corrosion and circular form has never been fully inves- by overheating of the plates. tigated. Fairbairn's experiments proved the weakness of lap-joints; but they of marine boilers may be assumed to comprise only two experiments with vary between the following limits, viz., elliptical flues, which furnish insufficient the length between 30 inches and 84 data for the construction of a formula inches, the diameter between 30 inches as attempted by Belpaire. and 42 inches, the thickness between inch and inch. The usual dimensions of furnace flues There are no records of tests made to determine the quality of the metal used in the construction of any of the experimental tubes. The behavior of the tubes under increasing pressures during the experiments has been recorded only in a single instance, viz., in the experiment made by the Leeds Forge Company, and then only imperfectly. While no reliable rules for proportioning and determining the strength of furnace flues can be deduced from the data furnished by the experiments heretofore made, these will be useful for checking the results obtained in future experiments-made with a special view toward determining the resistance of tubes representing the dimensions, construction and conditions of working of furnace flues as they occur in practice. When flues collapse during actual use in boilers the circumstances accompanying collapse and the immediate causes of the failure are seldom known sufficiently well to make such accidents the basis of accurate calculations. A first series of experiments should be made with a special view toward determining the influence of the several dimensions, viz, length, diameter and thickness, on the strength of flues. For this purpose each dimension should be varied at least three times within the above-given limits, under otherwise identical conditions of dimensions, quality of material and mode of construction. All these flues should be made, as nearly as possible, circular; they should be constructed in the manner which is most common in practice, and can be best relied upon to give uniform results; that is to say, riveted butt-joints with internal straps should be used for the longitudinal seams. The same material should be used in this whole series of experiments, and it should be of the kind commonly used in the construction of furnace flues, viz., Extra Firebox Iron, or a similar brand, the ductility and limit of elasticity of which have been carefully determined in the testing machine. A second series of experiments should be made with flues, circular in cross sec 5. RECOMMENDATIONS REGARDING FU-tion, and identical in every respect with TURE EXPERIMENTS ON THE RESISTANCE the flues used in the first series of exOF FURNACE FLUES TO COLLAPSE.periments, with the exception of either Exhaustive experiments on the resistance the mode of construction, or the mateof furnace flues to collapse should have rial, or both; that is to say, some flues |