tubes; the lower disc was braced by a ening effect produced by a deviation rod secured to it and to the lower cover from the circular form. of the cast-iron cylinder. From the foregoing experiments FairA few experiments were made in bairn deduced the following empirical which the cast-iron ends of the tubes formula, viz.: were not braced, in order to determine whether the tension produced in the p=9,675,600 (2) ld tubes by bracing them had any influence on their resistance to collapse. But the where the notation is the same as in forresults obtained were conflicting and mula (1), and l=length of tube in inches. were not regarded as conclusive by Fair In the year 1874 experiments were bairn. made at the Navy Yard, Washington, Some of the experimental tubes tested D. C., with two large cylindrical boiler by Fairbairn were thicker than 0.043 flues. (See Shock, “Steam Boilers,” p. inch, viz: 113). One tube (a) 184 inches in diameter The apparatus used for this purpose and 61 inches long, and consisting of a consisted of a cylindrical shell 63 inches single sheet with a lap-joint, was made in diameter, constructed of plate-iron of plate iron 0.25 inch thick. It col- inch thick; the experimental flues were lapsed at a pressure of 420 pounds. 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-fue (f) 761 inches long and 54 inches in iron 0.14 inch thick. One (6) had a lon- inside diameter, was made of 4 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 (6) 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 7 An iron flue (c), consisting of three inches wide and I 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 experi144 and 1411 inches, its extreme length mental flue (measured between the inner 604 inches; the ends were of sheet-iron edges of rivet holes in flanges) was 717 inch thick. The rivets, į inch in diam. inches. eter, were spaced 14 inches apart. This One of the rings of this flue collapsed tube collapsed at a à pressure of 125 at a pressure of 105 pounds. The pounds. bulged part was pressed out and shored An iron flue (d), 15 inches in diameter up, and the pressure was again applied. and 0.125 inch thick, consisted of three This time collapse took place in the rings connected by flanges with a plate- other ring at a pressure of 120 pounds. iron diaphragm between them. The The operation of forcing out and shorlength of the two outer rings was 21+ ing up the bulge was repeated several inches, that of the middle ring was 17 times, the tube becoming stiffer each inches. The rivets began to leak at 150 time. At the fifth trial collapse took pounds, and after being re-caulked, one place at a pressure of 186 pounds. It of the outer rings collapsed at 146 was found that the sheet which collapsed pounds. This experiment was rejected first was slightly less in thickness than by Fairbairn, who suspected that the inch, and that the flue was slightly oval, tube was originally defective. the larger diameter being 541 inches, and A steel flue (e), having the same di. the smaller diameter 53 inches. mensions and constructed in like manner A second flue (9) was made of inch as flue (d), collapsed at 220 pounds in boiler iron, the sheets having been accuthe middle ring. This flue was slightly rately gauged before fitting them. This elliptical, the diameters being 15 inches fue consisted likewise of two rings, 38 and 15% inches. inches and 39 inches long over all reTwo experiments were made with spectively, which were connected by elliptical tubes, which showed the weak- flanges with a ring 11 inch thick between VOL. XXIV. No. 3–15. 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 press in the narrow plate between the two ure of 130.6 pounde. welds, where the thickness was fully Experiments were made in March, 0.375' inch. It was remarkable how 1878, by the Leeds Forge Company, slight were the indications of weakness England, with a plain cylindrical and a preceding the collapse; a total deflection corrugated furnace flue, to test their rel- and permanent set of 0.04 inch at the ative strength when exposed to a col- point a being the only sign. lapsing pressure. (See “Engineering,” According to Fairbairn’s formula (2), Vol. XXV, pp. 245-260). this flue should have borne about 360 The plain cylindrical fiue (1) was made pounds pressure, while it actually bore of two iron plates ž 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 be. making the length 97 inches over all. ing merely stiffened by stout rings and The test vessel was a wrought-iron cy- otherwise left free to move. linder, about 3 inches larger inside than In addition to the foregoing experithe experimental flue, and strengthened ments, a few cases of the collapse of with five welded rings 5" X 3'', bored and tubes are recorded by different writers, shrunk on. The ends were formed by in which the conditions under which colcast-iron rings, tied to each other by long lapse took place were sufficiently well bolts outside the test vessel, and bored known to make the data valuable for to fit the turned ends of the flues, with comparison. a groove to receive a cupped leather Fairbairn gives two examples of new packing ring. This arrangement allowed boilers in which the flues became oval the flue to expand or contract freely. during the application of the hydraul On 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 2) were 42 inches in diameter, i 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 (2) 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 the Institution of Civil Engineers," Ses. 50 pounds at first, and of 25 pounds af- sion 1875–76, Part IV.). The flue (m) terward, and were taken off again after was old; originally f inch thick, it had each increase. At 200 pounds pressure been reduced to nearly to inch by corro 12.19 . FROM THE cy 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 col p=9,675,600 la lapsed part extending about one-half of the length of the flue. Fairbairn says that this “is the general An experiment is recorded in Julien formula for calculating the strength of et Bataille, “ Machines à Vapeur,” p. wrought-iron tubes subjected to external 240. The flue (n) had longitudinal and pressure, within the limits indicated by circumferential joints. Its diameter was the experiments; that is, provided that 7.87 inches, its length 276 inches, and its their length is not less than 1.5 feet, and thickness 0.157 inch. It collapsed at a not greater probably than 10 feet.” It pressure of 110 pounds. is, however, to be observed that the 3. FORMULA DEDUCED higher limit of length is arbitrarily fixed, FOREGOING EXPERIMENTS BY DIFFER since none of the experimental tubes exENT Writers.--Fairbairn thought that ceeded 61 inches in length. the results of his experiments warranted Fairbairn states tbat a closer approxithe conclusion that the resistance of mation to the experimental results is lindrical tubes to collapse varies in the given by the formula inverse ratio of their diameter and ť:10 a p=9,675,600 -0.002 (3) length. In deducing a formula from his lal t 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 2a. thickness is a constant quantity, and For elliptical flues the value is to be that the resistance of thin tubes to col. 6 lapse follows the same law as the resist- substituted for din formula (2), where a ance of thin iron plates to crumpling, is the greater, and b is the lesser semiwhich varies directly as a certain power of . of their thickness; the number indicat He recommends the use of the simpler formula ing this power has been found to lie between 2 and 3. to p=9,675,600 za Selecting twenty experiments with (4) 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 sev- that, by substituting the exponent 2 for for practical purposes. It is easily seen eral tubes, and that (with one exception, 2.19, the value of p will be increased for viz., the 8-inch tubes) the mean value of thicknesses less than one inch, and depld for each set of tubes of equal diam creased for thicknesses greater than one eter decreased as the diameter of the tubes increased, but not in a regular inch: The increase will amount to 70 ratio. The greatest discrepancy existed per cent. when t is 16 inch, 20 per cent. in the results of the experiments with when t is 3 inch, ando per cent. when is 1 inch. 12-inch tubes. Introducing the mean value of the ex. vations regarding Fairbairn's formula, R. Wilson makes the following obserpression pld into an equation of the viz.: pld form Fairbairn determines the “Its applicatlon is limited to wroughtP,1,d, iron tubes above a certain length, for it value of the exponent x from the three can easily be shown that, if we take the experiments marked (a), (b), (c), in Sec-crushing (crippling) strength of the ma 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 90,000 in feet at which the crushing (crippling) P= (6) strength of the tube and its collapsing (L+1)d strength, according to Fairbairn's rule, For lap-joints and for inferior workmanare equal, and any further reduction of ship the numerical factor may be rethe length will not give the increase of duced as low as 60,000. strength as it should do if the formula The rules of Lloyd's Register as well were correct. as those of the Board of Trade pre“ It is probable that with ordinary scribe further, that in no case the value tubes of the above lengths, a collapsing of P must exceed the amount given by pressure over one-half the crushing the following equation, viz.: (crippling) pressure will never be attained. 80000 P= “If Fairbairn's formula is worth any. (7) d thing at all, we shall gain nothing in collapsing resistance to hydraulic press-est working pressure in pounds per In formula (5), (6), (7), P is the highure by making § inch thick tubes in shorter lengths than 11 feet, seeing that and diameter in inches, L is the length square inch, and d are the thickness this is the length given by the rule (viz. of the flue in feet measured between the formula 2), at which the collapsing strengthening rings, in case it is fitted strength is half the crushing (crippling) with such. Formula (5) is the same as pressure The greater the thick- formula (4), with a ness of the tube in proportion to the di formula (4), with a factor of safety ameter, or the stiffer the tube the more is increased by 1; the influence which equal to . In formula (6) the length L will the strengthening rings tend to this addition has on the value of P is, bring the collapsing pressure up to the of course, greater for short tubes than crushing (crippling), pressure. A con- for long ones. sideration of the resisting powers of the umn V1 of the subjoined table, it will be strengthening rings themselves should also not be omitted in an investigation for formulæ (5) and (6) is by no means seen that the limit fixed by formula (7), of this matter at the high pressures we too low. are assuming. For a finch thick tube Grashof selected twenty-one of Fairstrengthening rings, at 3-feet lengths bairn's experiments (seventeen of which over the fire, still serve a very useful were on tubes 0.043 inch thicker, and four purpose in keeping the tube in shape in the event of overheating, and in restrict- deduced from them the following formthe event of overheating, and in restrict- on tubes from 5 to inch thick), and ing the amount of distortion in the event of collapse, and should always be ula, viz.: 2.315 employed here." (See Engineering, p=24,481,000 (8) June 2, 1876). Id 1.278 Fairbairn's formula (4) has been gen This formula agreed, however, better erally accepted as the basis of rules for ascertaining the strength of boiler flues with the results of the experiments with thin tubes than of those with thicker ones; In some cases, however, limits are fixed to its application by a supplementary formula, which gives a close agreement Grashof devised, therefore, a second formula of the form of equation (1). Lloyd's Register contains the following with the four experiments on tubes from formula for the strength of circular 3 inch to { inch thick, viz.: boiler flues, viz.: p=1,033,620 (9) (0.564 d 0.889? (5) Ld where the notation is the same as in form ula (2). (See “Theorie der Elasticitat The English Board of Trade pre- und Festigkeit,” von Dr. F. Grashof, p. scribes the following formula for circu- 328.) lar flues, when the longitudinal joints are Nystrom has deduced from Fairbairn's t 2.081 P=89,600 T-pdb experiments the following formula for wide, the total uniformly distributed load the collapsing strength of flues, viz.: on one are of such a strip is p 1 pounds, 4T ta when t is the length of the arc; and the p= (10) resistance to bending is given by the dL equation where p, t and d have the same meaning as in formula (2), L is the length in feet, ff (12) and T is the tensile strength of the metal λ in pounds per square inch. If to the Consequently the collapsing pressure 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 ing form, viz. : arcs into wbich the tube divides; and the strength of the tube must depend on the number of arcs into which the tube p=692,800 (10a) 20.50 divides. From the results of 19 experi. which is more convenient for comparison ments with thin tubes Unwin finds the with formula (9.) Nystrom considers the equation factor of safety I sufficient in apply 1=0.6375 20.45/0.08 (13) ing his formula. (See “A new Treat- which gives the best average value for ise on Steam Engineering,” by J. W. the length of the arcs into which the Nystrom, p. 106). tubes divided during a collapse. Formulæ (2), (5), (8), (9) and (10) have Unwin assumes that the metal of the the common defect that they make the flue is in the same condition as a straight collapsing pressure decrease indefinitely column of the length ad, subjected to a with increase of length and vice versa. compression of the intensity experiments an equation of a different = (14) form, which, reduced to English meas 2 ures, is as follows, viz. : where b is the length of a slice of the ť , d . p=5,358,150 + 41,906 = +1,323 ld d d to the investigation of the strength of where the notation is the same as in flues. According to Euler's theory we formula (2). Prof. W. C. Unwin says, have regarding this formula: “It is based to IE a certain extent on theoretical considera T= (15) 22 tions bearing on the probable limit of decrease of collapsing pressure with in- where I is the moment of inertia of the crease of length, and it no doubt repre- section of the column about an axis, sents very closely Fairbairn's experi- through its center of figure, and perpenments." dicular to the plane of bending; E is the Unwin, who had been associated with modulus of elasticity of the material; T Fairbairn in making the experiments is the greatest thrust consistent with described in section (2), has published a stability. For a rectangular section, re-investigation of them in the “Pro. bt3 ceedings of the Institution of Civil En I= then 12 gineers," Vol. XLVI., Session 1875–76, Part IV., of which the following is an n'Eb T= (16) abstract: 12 1? He shews first that in the experiments Putting E=28,500,000 we have with thin tubes, which had a uniform thickness of 0.043 inch, the originally cir bts cular tubes bent during collapse into T=23,440,000 (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 we get ratio of length to diameter decreased. 46,880,000 t p= (18) Considering a strip of a tube 1 inch dx |