tubes; the lower disc was braced by a rod secured to it and to the lower cover of the cast-iron cylinder. A few experiments were made in which the cast-iron ends of the tubes were not braced, in order to determine whether the tension produced in the tubes by bracing them had any influence on their resistance to collapse. But the results obtained were conflicting and were not regarded as conclusive by Fairbairn. Some of the experimental tubes tested by Fairbairn were thicker than 0.043 inch, viz: One tube (a) 183 inches in diameter and 61 inches long, and consisting of a single sheet with a lap-joint, was made of plate iron 0.25 inch thick. It collapsed at a pressure of 420 pounds. where the notation is the same as in formula (1), and l-length of tube in inches. In the year 1874 experiments were made at the Navy Yard, Washington, D. C., with two large cylindrical boiler flues. (See Shock, "Steam Boilers," p. 113). The apparatus used for this purpose consisted of a cylindrical shell 63 inches in diameter, constructed of plate-iron inch thick; the experimental flues were riveted to flanges within this shell. Two tubes, both 9 inches in diameter For the first experiment a cylindrical and 37 inches long, were made of plate- flue (ƒ) 764 inches long and 54 inches in iron 0.14 inch thick. One (b) had a lon- inside diameter, was made of inch gitudinal butt-joint with an outside cov-boiler iron. It consisted of two rings ering plate, and collapsed at a pressure connected by an interior butt-strap 72 of 378 pounds. The other one (b) had inches wide and inch thick. Each ring a longitudinal lap-joint, and bore only was formed of two plates with butt262 pounds. joints, having interior butt-straps 72 An iron flue (c), consisting of three inches wide and inch thick. The loncourses with lap-joints, the longitudinal gitudinal seams of the two rings broke seams breaking joint, was made of plates joint. All seams were double riveted. 0.125 inch thick. Its diameters were The unsupported length of the experi14 and 1411 inches, its extreme length mental flue (measured between the inner 60 inches; the ends were of sheet-iron edges of rivet holes in flanges) was 71 inch thick. The rivets, inch in diam. inches. eter, were spaced 11 inches apart. This One of the rings of this flue collapsed tube collapsed at a pressure of 125 at a pressure of 125 at a pressure of 105 pounds. The pounds. An iron flue (d), 15 inches in diameter and 0.125 inch thick, consisted of three rings connected by flanges with a plateiron diaphragm between them. The length of the two outer rings was 214 inches, that of the middle ring was 17 inches. The rivets began to leak at 150 pounds, and after being re-caulked, one of the outer rings collapsed at 146 pounds. This experiment was rejected by Fairbairn, who suspected that the tube was originally defective. A steel flue (e), having the same dimensions and constructed in like manner as flue (d), collapsed at 220 pounds in the middle ring. This flue was slightly elliptical, the diameters being 15 inches and 153 inches. 16 Two experiments were made with elliptical tubes, which showed the weakVOL. XXIV. No. 3-15. bulged part was pressed out and shored. up, and the pressure was again applied. This time collapse took place in the other ring at a pressure of 120 pounds. The operation of forcing out and shoring up the bulge was repeated several times, the tube becoming stiffer each time. At the fifth trial collapse took place at a pressure of 186 pounds. It was found that the sheet which collapsed first was slightly less in thickness than inch, and that the flue was slightly oval, the larger diameter being 54 inches, and the smaller diameter 53 inches. A second flue (g) was made of inch boiler iron, the sheets having been accurately gauged before fitting them. This flue consisted likewise of two rings, 38 inches and 39 inches long over all respectively, which were connected by flanges with a ring inch thick between them, as in the Adamson joint. This the flue began to come down like a blisflue was found to be perfectly cylindri- ter, about 5 inches in diameter, on the cal, having an internal diameter of ex- top of the flue at the point a of the origactly 54 inches. One of the rings col- inal prominence. The pressure began lapsed at a mean pressure of 133.3 to fall at first slowly, then quickly, until pounds, as indicated by three spring it reached 50 pounds, when the swelling gauges. The bulged part was shored had extended 24 inches by 12 inches by up and a second trial was made, when 18 inches deep. The collapse took place the other ring collapsed at a mean pressure of 130.6 pounds. Experiments were made in March, 1878, by the Leeds Forge Company, England, with a plain cylindrical and a corrugated furnace flue, to test their relative strength when exposed to a collapsing pressure. (See "Engineering," Vol. XXV, pp. 245-260). in the narrow plate between the two welds, where the thickness was fully 0.375 inch. It was remarkable how slight were the indications of weakness preceding the collapse; a total deflection and permanent set of 0.04 inch at the point a being the only sign. According to Fairbairn's formula (2), this flue should have borne about 360 The plain cylindrical flue (h) was made pounds pressure, while it actually bore of two iron plates inch thick, of un- but little more than one-half that pressequal width, the narrower plate being ure. Although the tube was not accuabout 24 inches wide. These plates rately circular, the irregularities of form were lap-welded, the welded parts being were not greater than are common in hammered down to the thickness of the welded flues, and were less than in ordiadjacent portions of the plates. The nary riveted lap-jointed flues. It may be actual thickness of the plates varied assumed that the close proximity of the from 0.36 to 0.40 inch. The flue had an two welds was a source of weakness. It outside diameter of about 37 inches, is also probable that the flue would have and a length of 84 inches, with a turned resisted a greater pressure, if the ends. collar in addition at each end, welded on, had been rigidly secured, instead of bemaking the length 97 inches over all. ing merely stiffened by stout rings and otherwise left free to move. The test vessel was a wrought-iron cylinder, about 3 inches larger inside than the experimental flue, and strengthened with five welded rings 5"x3", bored and shrunk on. The ends were formed by cast-iron rings, tied to each other by long bolts outside the test vessel, and bored to fit the turned ends of the flues, with a groove to receive a cupped leather packing ring. This arrangement allowed the flue to expand or contract freely. In addition to the foregoing experiments, a few cases of the collapse of tubes are recorded by different writers, in which the conditions under which collapse took place were sufficiently well known to make the data valuable for comparison. Fairbairn gives two examples of new boilers in which the flues became oval during the application of the hydraulOn testing the flue with straight ic test pressure. (See "Philosophical edges, it was found to have prominences Transactions," 1858). The flues (k and from inch to inch high. Two of ) were 42 inches in diameter, inch these points (marked a) at a distance of thick, and 35 feet and 25 feet long reabout four-tenths the length of the tube spectively. The longer flue (k) became from one end, were selected for measur- oval at a pressure of 97 pounds, and the ing the horizontal and vertical diameters shorter flue (1) at a pressure of 127 which were found to be 36.65 inches and pounds. 37.20 inches respectively. A slight dif- An experiment made by Mr. Alfrey ference in the horizontal and vertical of the firm Humphreys, Tenant & Co., diameters was also found near the other is cited by W. C. Unwin in a paper on end, where points were selected for the "Resistance of Boiler flues to Colmeasurements during the trial. The lapse," contained in the "Proceedings of pressures were raised by increments of 50 pounds at first, and of 25 pounds afterward, and were taken off again after each increase. At 200 pounds pressure the Institution of Civil Engineers," Session 1875-76, Part IV.). The flue (m) was old; originally inch thick, it had been reduced to nearly inch by corro sion. There were four longitudinal lap- tion (2). The three values of a thus joints in the circumference of the flue, found were 2.23, 2.14 and 2.16; and takand four circumferential lap-joints in its ing 2.19 as the mean of these values, he length. The diameter of the flue was uses this number as the exponent of tin 33 inches, its length 360 inches, its thick- his equation. In this manner he gets ness 0.34 inch. It collapsed at a press- formula (2), viz.: ure of 99 pounds over an arc of about one-eighth of its circumference, the collapsed part extending about one-half of the length of the flue. An experiment is recorded in Julien et Bataille, "Machines à Vapeur," p. 240. The flue (n) had longitudinal and circumferential joints. Its diameter was 7.87 inches, its length 276 inches, and its thickness 0.157 inch. It collapsed at a pressure of 110 pounds. 3. FORMULA DEDUCED FROM THE FOREGOING EXPERIMENTS BY DIFFER ENT WRITERS.-Fairbairn thought that the results of his experiments warranted the conclusion that the resistance of cy lindrical tubes to collapse varies in the p=9,675,600 12.19 ld Fairbairn says that this "is the general formula for calculating the strength of wrought-iron tubes subjected to external pressure, within the limits indicated by the experiments; that is, provided that their length is not less than 1.5 feet, and not greater probably than 10 feet." It is, however, to be observed that the higher limit of length is arbitrarily fixed, since none of the experimental tubes exceeded 61 inches in length. Fairbairn states that a closer approximation to the experimental results is given by the formula p=9,675,600 12.19 d -0.002 (3) t inverse ratio of their diameter and length. In deducing a formula from his experiments he starts with the assump- which allows for a slight deviation from tions that the product of the collapsing the circular form owing to the thinness pressure, per square inch, into the length of the plates. and diameter of similar tubes of equal thickness is a constant quantity, and that the resistance of thin tubes to col lapse follows the same law as the resistance of thin iron plates to crumpling, which varies directly as a certain power of their thickness; the number indicating this power has been found to lie between 2 and 3. 2a' For elliptical flues the value is to be b substituted for d in formula (2), where a is the greater, and b is the lesser semiaxis of the ellipse. He recommends the use of the simpler formula p=9,675,600 ld (4) Selecting twenty experiments with tubes having a thickness of 0.043 inch, saying that for thick tubes of consideraFairbairn deduces from their results a able diameter and length, this formula mean value for the expression pld. It is, however, to be observed that this value may be regarded as sufficiently accurate varied in many cases greatly for the several tubes, and that (with one exception, viz., the 8-inch tubes) the mean value of pld for each set of tubes of equal diameter decreased as the diameter of the tubes increased, but not in a regular ratio. The greatest discrepancy existed in the results of the experiments with 12-inch tubes. for practical purposes. It is easily seen is 1 inch. inch, 20 per cent. 0 per cent. when t P= 90,000t (6) terial at 12 tons per square inch, the welded, or made with riveted butt-straps, number 15 multiplied by the thickness in viz.: inches of the tube will give the length in feet at which the crushing (crippling) strength of the tube and its collapsing strength, according to Fairbairn's rule, are equal, and any further reduction of the length will not give the increase of strength as it should do if the formula pressure For lap-joints and for inferior workmanship the numerical factor may be reduced as low as 60,000. The rules of Lloyd's Register as well. as those of the Board of Trade prescribe further, that in no case the value of P must exceed the amount given by the following equation, viz.: 8000t P = d (7) and diameter in inches, L is the length square inch, t and d are the thickness of the flue in feet measured between the strengthening rings, in case it is fitted with such. Formula (5) is the same as formula (4), with a factor of safety equal to t. In formula (6) the length L is increased by 1; the influence which this addition has on the value of P is, of course, greater for short tubes than for long ones. By an inspection of column VI of the subjoined table, it will be for formulæ (5) and (6) is by no means seen that the limit fixed by formula (7), "If Fairbairn's formula is worth anything at all, we shall gain nothing in In formula (5), (6), (7), P is the highcollapsing resistance to hydraulic press-est working pressure in pounds per ure by making inch thick tubes in shorter lengths than 11 feet, seeing that this is the length given by the rule (viz. formula 2), at which the collapsing strength is half the crushing (crippling) The greater the thickness of the tube in proportion to the diameter, or the stiffer the tube the more will the strengthening rings tend to bring the collapsing pressure up to the crushing (crippling) pressure. A consideration of the resisting powers of the strengthening rings themselves should also not be omitted in an investigation of this matter at the high pressures we are assuming. For a 3 inch thick tube strengthening rings, at 3-feet lengths over the fire, still serve a very useful purpose in keeping the tube in shape in the event of overheating, and in restrict-on tubes from to inch thick), and deduced from them the following forming the amount of distortion in the ula, viz.: event of collapse, and should always be employed here." (See Engineering, June 2, 1876). too low. Grashof selected twenty-one of Fairbairn's experiments (seventeen of which were on tubes 0.043 inch thicker, and four p=24,481,000 t2.315 ld 1.278 (8) where p, t and d have the same meaning as in formula (2), L is the length in feet, and T is the tensile strength of the metal in pounds per square inch. Consequently the collapsing pressure If we assign to T the value 50,000, and express the length of the flue in necessary to produce bending varies ininches, equation (10) assumes the follow-versely as the square of the length of the arcs into which the tube divides; and ing form, viz. : the strength of the tube must depend on the number of arcs into which the tube divides. From the results of 19 experiments with thin tubes Unwin finds the t2 p=692,800 105d (10a) which is more convenient for comparison with formula (9.) Nystrom considers the equation factor of safety sufficient in applying his formula. (See "A new Treatise on Steam Engineering," by J. W. Nystrom, p. 106). Formula (2), (5), (8), (9) and (10) have the common defect that they make the collapsing pressure decrease indefinitely with increase of length and vice versa. M. Love has deduced from Fairbairn's experiments an equation of a different form, which, reduced to English measures, is as follows, viz. : where the notation is the same as in formula (2). Prof. W. C. Unwin says, regarding this formula: "It is based to a certain extent on theoretical considerations bearing on the probable limit of decrease of collapsing pressure with increase of length, and it no doubt represents very closely Fairbairn's experiments." Unwin, who had been associated with Fairbairn in making the experiments described in section (2), has published a re-investigation of them in the "Proceedings of the Institution of Civil Engineers," Vol. XLVI., Session 1875-76, Part IV., of which the following is an abstract: λ=0.6375 70.45 70.08 (13) which gives the best average value for the length of the arcs into which the tubes divided during a collapse. Unwin assumes that the metal of the flue is in the same condition as a straight column of the length nd, subjected to a compression of the intensity where b is the length of a slice of the flue, and d its diameter. He then applies the laws of resistance of long columns to the investigation of the strength of flues. According to Euler's theory we have He shews first that in the experiments with thin tubes, which had a uniform Putting E=28,500,000 we have thickness of 0.043 inch, the originally circular tubes bent during collapse into we get bts T=23,440,000 ; (16) (17) figures consisting of acres of alternately and combining equations (14) and (17), convex and concave curvatures, and that the number of arcs increased as the ratio of length to diameter decreased. Considering a strip of a tube 1 inch p= 46,880,000 t (18) |