a p=id 20.18 Introducing into this equation the value 2.35 of t=0.043, we find p=15,547,000 (24) 20.9 / 1.16 (19) In order to make the latter formula Npd convenient for arithmetical calculations, Combining equation (19) with equation he gives it the form (13), we obtain for the collapsing press p=15,547,000 (240) ure of tubes 0.043 inch thick, the following formula, viz.: where a ß y are variable co-efficients, 9170 p= (20) the values of which, corresponding to 20.9 d 1.16 certain values of t, I and d, are given by Fairbairn's formula for the same tubes, is him in a table adjoined to his investiga tion. 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, Imax. = 6.7 d’ similar experiments leads to more relia. 2.22 ble results than the mixing up of heteronearly; Imin. = 4468, Tne two geneous experiments to obtain average d values of constants. In applying these limits coincide when t= For greater formula to practical cases, it ought to be 19 values of t the formule cease to be applic- the circular form may greatly affect the borne in mind that slight deviations from able. Assuming that the preceding investi- of the flue." value of 1, and much reduce the strength gations indicate that the collapsing press Theodore Belpaire has made an attempt ure of tubes is given by an equation of to develop a formula for the strength of the form flues from experimental data, from which the variable and uncertain element of d strength due to circularity of form has and, taking the experiments on thin tubes that flues derive their main strength from been eliminated. He starts with the idea as a starting point of the comparison, the rigid fastenings of their ends. Con: Unwin deduces values of the factor c and the exponent n from a number of experi- sidering the flexure of a narrow strip of ments with thicker tubes, combining the a tube cut parallel to its axis, he finds that the bending forces are so insignifiresults obtained with tubes of similar construction. cant that they may be neglected without From two experiments with tubes his investigation on the equation giving sensible error; and he bases, therefore, made with a longitudinal lap.joint, the greatest shearing forces which exist [marked (a) and (6), in section 2], he at the rigidly secured ends of the loaded deduces the formula strip, viz.: (22) SE (26) 2t From one experiment with a tube hav The highest value of S consistent with ing a longitudinal butt-joint, (marked (6) stability is to be deduced from experiin section 2], he gets the formula mental data. Belpaire rejects all exper (2.21 iments with circular flues, and uses those (23) made with elliptical ones, because in the latter cases the relation existing between From five experiments with tubes hav- the strength and the form of the transing longitudinal and circumferential verse section can be deduced with some joints [marked (c), (k), (l), (m) and (n), degree of certainty. in section 2], he derives the formula Calling D the greater and X the lesser p=€70.9 (1.16 p=9,614,0000.9 1.16 t diameter of the ellipse, he represents its all tubes derive more or less additional eccentricity by strength from these causes, he considers D-X one-quarter a sufficiently large factor of D (27) safety in proportioning flues by means of S is evidently a function of the eccentric formula (33). (See “Note sur la résistity, and may be expressed with sufficient ance des tubes pressées de l'extérieur,” accuracy by an eyuation of the form par Théodore Belpaire, in " Annales du Génie civil," Mars, 1879). (28) Equation (33) may be written as folSince the resistance to flexure ap- lows, viz: proaches zero, and consequently. S=0, when E=1, and since, on the other hand, p=3,427,152 1-16.6 ld d S=a maximum, when e=0, the equation (28) can be put under the following The factor (1–16.6 becomes zero, and form, viz.: S=A (1-e?.) (29) consequently p=0, when 16.6 S (and consequently A) is likewise a , t 16.6 t. function of This indicates that the formula and an approximate value is, at best, applicable only within certain d' for A may be deduced from an equation limits, which, however, have not been of the form determined. D. K. Clark, in his “Manual of Rules," A=m+n (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 m=0. Introducing this value into use in boilers. These flues varied from equation (30), and combining equations 24 to 60 inches in diameter, and from (29) and (30) we get the equation ito sinch in thickness. They con sisted of rings of plates riveted together, seams, but all of them unfortified by intermediate Belpaire deduces values for n and , flanges or strengthening rings. At the from two experiments by Fairbairn on collapsing pressures the flues experienced elliptical tubes. Introducing the values compressions ranging from 1.53 to 2.17 thus found into equation (31), and mak. tons, or a mean compression of 1.82 tons ing e=0, he gets the following expres- per square inch of section. From these sion for S applicable to tubes of a circu- data Clark deduced the following formlar cross-section, viz.: ula "for the average resisting force of common boiler flues,” viz : (32) (50,000 -500 (34) d Introducing this value of Sinto formula (26), we get is the collapsing pressure in t" pounds per square inch, and d and t are. p=3,427,152 ld Id" (33) the diameter and thickness expressed in inches. Applying this formula to a number of It is assumed that the flues are not examples of tubes collapsed under known strengthened by rings. The influence conditions, Belpaire finds that the actual of length on the strength of flues Clark collapsing pressures were from 1.05 to calls an uncertain element; it is, how4.15 times as great as the pressures given ever, a very important one, as proven by by formula (33). These variations he Fairbairn's experiments, and for this reaascribes to the influence of the uncertain son the formula is not generally applicaelements of stiffness, circularity of form ble. and homogeneity of material, which, he The following table contains the data avers, should not be allowed to influence of all experiments on the collapse of a formula of this kind. Since, however, flues having thickness of one-eighth inch s={~(á)+2 () }(1-6) (31) with one or two longitudinal seama S=1,713,570 (á) –28,446,200 p=t where p t3 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 Experiments Nos. 4 and 5 were rethe different formulæ, of which an ac- jested by Fairbairn and Unwin, but count has been given. In comparing the perhaps not on sufficient grounds. Nyscalculated with the actual collapsing trom's formula (11) agrees quite well with pressures, for the purpose of estimating the results of these experiments. the reliability of the several formula, it Belpaire's formula (33) gives in only is, of course, necessary in each case not one case a close agreement with the to take into consideration those experi- actual results of the experiments; and the ments from which the formula was de- degree of agreement varies widely in the duced. other cases. In Fairbairn's formula (2) the index number of the power of t was deduced 4. INSUFFICIENCY OF THE ABOVE-DE. from the results of experiments Nos. 1, 2 SCRIBED EXPERIMENTS.-It will be noand v. Leaving these experiments out ticed that many of the formulæ given in of consideration, only oue experiment, the previous section, though deduced viz., No. 11, shows a tolerably close mostly from the same set of experiagreement between the actual and the ments, and being similar in form, assign calculated collapsing pressures; all other very different values to the influence of experiments give too large a value for length, diameter and thickness of tubes flues having a length less than 60 inches, on their resistance to collapse, and that and too small a value for flues having a most of them give very discordant relength of 276 inches or more. sults. Fairbairn's approximate formula (4) The differences in the two formula, gives in every case a larger value for the (8) and (9), deduced by Grashof from collapsing pressure than formula (2), as Fairbairn's experiments, illustrate plainly previously explained. It is easily seen the fact that the numerous experiments that the close agreement between the on very thin tubes are of relatively little actual and calculated results in the case value for determining the strength of of experiments Nos. 9 and 10 is purely flues used in steam boilers. In fact, all accidental. those formule which are based princiGrashof's formula No. (9) gives in pally upon the experiments with tubes every case much too high a value of p, 0.043 inch thick, are very defective. except in experiments Nos. 1, 2, 5 and This remark applies specially to Fair6, from which the formula was deduced. bairn's formula (2), to Love's formula Nystrom’s formula (10) was deduced (11), and also to Unwin's formulæ (22), from Fairbairn's experiments, but it is (23) and (24). Unwin recognized the not stated which of them were used. As fact that it was hazardous to apply to in no case the tensile strength of the thick tubes the rules relating to the inmetal of which the experimental flues fluence of length and diameter on the were made has been recorded, an aver- strength deduced from very thin tubes; age value of T=50,000 in the case of but he was compelled to do so by the iron flues, and of T=70,000 in the case paucity of available experiments. He of the steel flue, has been assumed. On derived the numerical factor and the inthe whole, this formula gives a closer dex number of t in formula (22) from agreement of the calculated with the two experiments, those in formula (23) actual collapsing pressures in experiments from one experiment, and those in foron flues of every description than any of mula (24) from five experiments, on the other formula. tubes from } inch to ź inch thick. There Love's formula (11) gives in every case are not a sufficient number of experitoo large a value, probably because it ments extant to test fully the value of was deduced mainly from Fairbairn's his formula. 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 AJTUAL AND CALCULATED COLLAPSING PRESSURES OF FLUES. IV. V. VII. VIII. IX. X. XI. XII XIII, XIV. XV. XVI. XVII. XVIII. XIX. XX. XXI. Number of experiment. Length in inches. P22, P22 P10 р P, P11 ecc P 1 61 18.75 0 25 3.5 15,750 420 406* 0.97 529 1.26 420* 1.00 296 0.70 450 1.07 331* 0.79 146 0.35 Longitudinal lap-joint, Fair bairn's experiment [a.] 2 37 9 0.14 4.1 12,150 378 392* 1.04 569 1.51 319* 0.85 248 0.66 427 1.13 378* 1.00 150 0.40 Longitudinal butt joint, Fair bairn's experiment [6.] 3 37 9 0.14 4.1 8,421 262 392 1.49 569 2.17 319 1.22 248 0.95 427 1.60 359* 1.37 150 0.57 Longitudinal lap-joint, Fair bairn's experiment [6.] bairn's experiment [a.] Fairbairn's experiment [e.] Fairbairn's experiment [C] 7 420 42 0.375 10.0 5,433 97 64 0.65 77 0.79 160 1.65 113 1.17 195 2.01 0.91 23 0.24 Longitudinal & transverse joints [k.] 8 300 42 0 375 7.1 7,112 127 90 0.71 108 0.85 194 1.53 134 1.05 212 1.67 120* 0.94 32 0.26 Longitudinal & transverse joints [1.] [m.] [n.] 0.25 0.67 14, 105 130.6 239 1.84 311 2.38 225 1.72 133 1.02 227 1.74 175 1.34 101 0.77 Longitudinal butt-joints [..] 13 84 37.25 0.375 2.25 9,933 200 361 1.80 435 2.17 442 2.21 285 1.42 414 2.07 307 1.54 128 0.64 Longitudinal butt-joints (weld ed) [h.] * Experiment used in deducing formula, 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 (9), 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 duce and 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 usual dimensions of furnace flues 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 1 There are no records of tests made to inch and inch. determine the quality of the metal used A first series of experiments should be in the construction of any of the experi- made with a special view toward determmental tubes. ining the influence of the several diThe behavior of the tubes under in-mensions, viz, length, diameter and creasing pressures during the experi- thickness, on the strength of flues. For ments has been recorded only in a single this purpose each dimension should be instance, viz., in the experiment made by varied at least three times within the the Leeds Forge Company, and then above-given limits, under otherwise only imperfectly. identical conditions of dimensions, qualiWhile no reliable rules for proportion- ty of material and mode of construction. ing and determining the strength of All these flues should be made, as nearly furnace flues can be deduced from the as possible, circular; they should be condata furnished by the experiments here. structed in the manner which is most tofore made, these will be useful for common in practice, and can be best rechecking the results obtained in future lied upon to give uniform results; that experiments-made with a special view is to say, riveted butt-joints with intertoward determining the resistance of nal straps should be used for the longitubes representing the dimensions, con- tudinal seams. The same material should struction and conditions of working of be used in this whole series of experifurnace flues as they occur in practice. ments, and it should be of the kind comWhen flues collapse during actual use in monly used in the construction of furboilers the circumstances accompanying nace flues, viz., Extra Firebox Iron, or a collapse and the immediate causes of the similar brand, the ductility and limit of failure are seldom known sufficiently elasticity of which have been carefully well to make such accidents the basis of determined in the testing machine. accurate calculations. A second series of experiments should be made with flues, circular in cross sec5. 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 |