= of 25o = 0:4142, hence with radius }, as in our case, we have a, b 0·2071, and a, c = 0·4142 inch. The pressure on a, c will act direct as a tendency to rupture on the points B, B, the force being 0.4142 x 100 41.42 lbs.; but the pressure on c, e and a, g will act obliquely. Thus the strain on c, e will of course be 41.42 lbs., as on a, c, but it will act in the direction of the arrow x, and must be resolved into two equivalent forces, one in the direction of the arrow y, which being at right angles to B, B, will tend to rupture on those points; the other in the direction of the arrow z being parallel to B, B will have no effect. By the well-known parallelogram of forces, Fig. 20 making the diagonal D = 41.42 lbs., we have two equivalent strains, the direction and force of which are given by the two sides of the parallelogram E, F, each 29.29 lbs., F being a direct tensile strain on the points B, B in Fig. 19. Of course the side a, g will give 29.29 lbs. also, and the combined strain will be from a, g 29:29; a, c = 41.42; and c, e = 29.29 lbs., or 29.29 + 41.42 +29.29 100 lbs., being precisely the same as with the tube, Fig. 18, 1 inch square. Calculating in this way with a polygon of any number of sides we should obtain the same result, and a circle being regarded as a polygon with an infinite number of sides, we thus find that the strain on a cylindrical tube is the same as on a square one of the same dimensions. From this it follows that the strength of a cylinder of thin plate, such as an ordinary boiler, is simply and directly proportional to the thickness, and inversely as the diameter. (57.) “ Lap-welded Tubes." —Say that we require the strength of a small boiler 24 inches diameter, -inch plate, with welded joint, made of Staffordshire plates whose tensile strength, namely, that of a solid plate, is 20 tons per square inch. By Mr. Bertram's experiments at Woolwich the strength of a lap-welded joint may be taken at 65 per cent. of that of the solid plate: hence 20 x .65 = 13 tons, or 29,120 lbs. per square inch. In our case rupture strains } a square inch (or 4 inch at each side); hence 29120 x } = 14560 lbs., which on 24 inches gives 14560 -- 24 = 607 lbs. per square inch = bursting pressure; with 6 for the Factor of safety (78) we obtain 607 = 6 = say 100 lbs. per square inch, safe or working pressure. From this we have the general rules : (58.) For welded boilers: P = 58200 x t = d. (59.) p = 9700 x = d. In which t thickness of plate in inches; d inside diameter in inches; P = the bursting pressure, and p = the safe working pressure in lbs. per square inch: thus for the 21-inch boiler we have considered, the rule gives 9700 1 = 24 101, say 100 lbs. per square inch working pressure, as before. (60.) “Steam-boilers with Riveted Joints."-Staffordshire plates are now so extensively used for boilers, that it will be expedient to take them as a basis for general rules, although, as shown by Table 5, their strength is inferior to the mean of British plateiron, and still more inferior to Yorkshire iron. We have shown in (44) that the pitch of rivets, and thereby the general proportions of joints in steam-boilers, is governed by the pressure of steam as affecting the tendency to leakage, irrespective of strength to resist bursting. (61.) For the purpose of fixing general proportions, we may take as a standard” case the working pressure of 50 lbs. per square inch, the proportions due to which will suffice for all lower pressures; and also with sufficient accuracy for practical purposes up to say 70 or 80 lbs. per square inch. The proportions for higher pressures should be found by special calculation (68) (76). We have first to find the space between rivet-holes with the different thicknesses of plate for 50-1b. steam by Table 13; taking the nearest pressures in that Table we obtain col. 6 in Table 14. Thus, for 3-inch plate we have for 50-lb. steam the space 1*6; Table 14 gives 1. rivets, as in col. 3 ; hence the pitch = 11 + 1 = 2 inches, col. 4; the ratio of 2 the metal between holes to the solid part of the plate = 11: • 2, or 21 : 32 = .656, as in col. 7. The apparent strength in single-riveted joints of Staffordshire plates being 34,110 lbs. per square inch by col. 7 of Table 5, that on the solid part of the plate = 34110 x .656 22380 lbs., as in col. 8 of Table 14. 11 e of 25o = 0·4142, hence with radius }, as in our case, we have a, b = 0.2071, and a, c = 0·4142 inch. The pressure on a, c will act direct as a tendency to rupture on the points B, B, the force being 0.4142 x 100 41.42 lbs.; but the pressure on c, e and a, g will act obliquely. Thus the strain on c, e will of course be 41.42 lbs., as on a, c, but it will act in the direction of the arrow x, and must be resolved into two equivalent forces, one in the direction of the arrow y, which being at right angles to B, B, will tend to rupture on those points; the other in the direction of the arrow z being parallel to B, B will have no effect. By the well-known parallelogram of forces, Fig. 20 making the diagonal D = 41.42 lbs., we have two equivalent strains, the direction and force of which are given by the two sides of the parallelogram E, F, each 29. 29 lbs., F being a direct tensile strain on the points B, B in Fig. 19. Of course the side a, g will give 29.29 lbs. also, and the combined strain will be from a, g = 29.29; a, c = 41.42; and c, e = 29.29 lbs., or 29.29 + 41.42 +29.29 = 100 lbs., being precisely the same as with the tube, Fig. 18, = 1 inch square. Calculating in this way with a polygon of any number of sides we should obtain the same result, and a circle being regarded as a polygon with an infinite number of sides, we thus find that the strain on a cylindrical tube is the same as on a square one of the same dimensions. From this it follows that the strength of a cylinder of thin plate, such as an ordinary boiler, is simply and directly proportional to the thickness, and inversely as the diameter. (57.) “ Lap-welded Tubes.”—Say that we require the strength of a small boiler 24 inches diameter, l-inch plate, with welded joint, made of Staffordshire plates whose tensile strength, namely, that of a solid plate, is 20 tons per square inch. By Mr. Bertram's experiments at Woolwich the strength of a lap-welded joint may be taken at 65 per cent. of that of the solid plate: hence 20 x .65 13 tons, or 29,120 lbs. per square inch. In our case rupture strains } a square inch (or inch at each side); hence 29120 x } = 14560 lbs., which 1 on 24 inches gives 14560 -- 24 = 607 lbs. per square inch = = bursting pressure; with 6 for the Factor of safety (78) we obtain 607 - 6 = say 100 lbs. per square inch, safe or working pressure. From this we have the general rules : (58.) For welded boilers: P = 58200 xt; d. (59.) p = 9700 xt; d. In which t = thickness of plate in inches; d = inside diameter in inches; P = the bursting pressure, and p = the safe working pressure in lbs. per square inch: thus for the 24-inch boiler we have considered, the rule gives 9700 x = 24 = 101, say 100 lbs. per square inch working pressure, as before. (60.) “Steam-boilers with Riveted Joints.”—Staffordshire plates are now so extensively used for boilers, that it will be expedient to take them as a basis for general rules, although, as shown by Table 5, their strength is inferior to the mean of British plateiron, and still more inferior to Yorkshire iron. We have shown in (44) that the pitch of rivets, and thereby the general proportions of joints in steam-boilers, is governed by the pressure of steam as affecting the tendency to leakage, irrespective of strength to resist bursting. (61.) For the purpose of fixing general proportions, we may take as a “standard” case the working pressure of 50 lbs. per square inch, the proportions due to which will suffice for all lower pressures; and also with sufficient accuracy for practical purposes up to say 70 or 80 lbs. per square inch. The proportions for higher pressures should be found by special calculation (68) (76). We have first to find the space between rivet-holes with the different thicknesses of plate for 50-1b. steam by Table 13; taking the nearest pressures in that Table we obtain col. 6 in Table 14. Thus, for 3-inch plate we have for 50-lb. steam the space 118; Table 14 gives it rivets, as in col. 3 ; hence the pitch = 1 + 11 = 2 inches, col. 4; the ratio of 16 +1 the metal between holes to the solid part of the plate = 11: 2, or 21 ; 32 = .656, as in col. 7. The apparent strength in single-riveted joints of Staffordshire plates being 34,110 lbs. per square inch by col. 7 of Table 5, that on the solid part of the plate 31110 x 656 22380 lbs., as in col. 8 of Table 14. 6 - TABLE 14.-Of the PROPORTIONS of RIVETED JOINTA in WROUGHT-IRON PLATES, for Steam-boilers, with 50-lb. Steam. Double Riveted. Diameter of Strain on Solid Part of Plate', in Los per Square Inch. Thickness. Pitch. Space Working Ratio of to Un. Plate. Single By Practical Sizes. Single Double Lap. Between |