3o8 



NA TURE 



[July 24, 1890 



thus determining /'« - i ; a knowledge of ^'« — j is required 

 when we come to the determination of the amount of shrinkage 

 necessary to produce pn — y 



Proceeding successively in this manner, we finally obtain — 





. . (iii.) 

 . . (ii.) 

 . . (i.) 



thus determining /o, the maximum allowable powder pressure in 

 the gun, for maximum working values of t^, Z^, t^ . . . in -ii 

 and these are the fundamental equations employed in gun- 

 making. 



(20) With no shrinkage, or a homogeneous gun, the maximum 

 allowable powder pressure would be reduced to 



so that we perceive the advantage of the shrinkage in strengthen- 

 ing the gun. 



(21) In Fig. 5 the dimensions are taken from the American 

 " Notes on the Construction of Ordnance," No. 31, by Lieut. 

 Rogers Birnie, slightly altered to round numbers ; the diameter 

 of the powder-chamber of the 8-inch gun is supposed to be 10 

 inches ; so that r^ = 5 ; and we put r^ = 7, ^2 = il, rj = 13, 

 ;-4 = 16; instead of 475, 7, ii, I3"i5, and 1575, as given in the 

 Note 31. 



Now, solving equations (25), (iv.), (iii.), (ii.), (i.) with/j = o, 

 .^3 = f^ = tj^ = 18, (q = 15, we shall find— 



Ps 



16^ - 13^- 

 1 6-^ + if 



31, 



14-3; 



13'- 11^ 



13- + 



2(18 -f 37) + 37 = 7-3. ^'3= 14-4 



7-3; 



■■ 4-3: 



Thus, the maximum allowable powder pressure in the chamber 

 of this gun is nearly 29 tons per square inch ; so that if the 

 pressure is limited to 1 7, the gun has- a factor of safety 

 29-M7= 17. 



Joining the tops of the ordinates for/,, and/i, ^^ and f\, &c., 

 by Barlow's curves, we have the graphical representation of the 

 maximum allowable firing stresses of this gun ; in which it must 

 be noticed that the area of the rectangle, p^r^ = 143 "5, is equal 

 to the area of all the circumferential tension curves bounded by 

 the jagged edge tijt\t~J:\. . . . 



(22) With a powder pressure p^ = 287 (tons on the square 

 inch) the powder stresses will be given by 



^ '^-^-^^^ 287 = 34-9, 



Pi=Po' 



16- 



and 



'5- - 16- - ' 

 ^1 = ^0 - A + A = i9'3 - ^'1 

 A = 3'5. ^2 = 97 = ^'"; 

 A=i-o, ^3 = 7-2 = ^'3; 



A = o. 



6-2. 



Subtracting these powder stresses from the firing stresses, we 

 are left with the initial stresses in the gun in a state of repose, 

 represented in Fig. 6, and given by 



(23) The data to which the gun-maker works are, first, the 

 calibre of the gun ; and secondly, the maximum powder pressure 

 to be expected at any point of the bore ; from these data, and 

 the quality of the steel at his command, and also from the 



NO. 1082, VOL. 42] 



capacity of his machinery in producing and shaping the various 

 pieces, the gun-maker proceeds to calculate the requisite thick- 

 ness and number of the coils, arranged so that the maximum 

 working tension shall not exceed certain practical limits laid 

 down (18 tons per square inch in the coils, and 15 in the tube). 



Thus, suppose he is called upon to design the cross-section of 

 a gun over the powder-chamber, 10 inches in diameter, to stand 

 a pressure of 20 tons per square inch. 



He will generally take a factor of safety, say 2, and allow for 

 double the pressure, so that he puts/o = 40, and then i^ = 15. 



He has r^ given as 5 inches, and now r^ is settled by the 

 manufacture of the solid steel block or log, which is bored out 

 to form the inner tube A ; and now he can calculate p^ and t\. 



Practical considerations of manufacture decide the thickness 

 and external radius 7-^ to be given to the jacket B ; and now, 

 knowing r^, ;'2, /i, and t^ = 18, he can calculate /o and i'^. 



Similar practical metallurgical and manufacturing considera- 

 tions decide the most suitable thickness for the hoops c, D, . . . ; 

 and when he finds the radial pressure has become zero (or 

 negative) the gun-maker knows that he has given his gun 

 sufficient thickness and strength. 



(24) A rule, suggested by Colonel Gadolin, was originally found 

 convenient, by which the radii of the coils were made to increase 

 in geometrical progression ; this rule, though useful when guns 

 were formed of a steel tube strengthened with wrought-iron 

 hoops, is obsolete now that steel is used throughout ; it was, 

 however, formerly employed as a first approximation in the 

 tentative solution of the problem. 



T/ie Longitudinal Stress in the Gun. 



(25) So far we have not yet taken into account the distribu- 

 tion of longitudinal tension in the gun ; and it must be confessed 

 that no satisfactory rigorous theory exists at present for the 

 determination. 



Practically it is usual to take the longitudinal tension as 

 uniform across a cross-section, and as due to the powder- 

 pressure in the bore, treated as a closed vessel, closed at one 

 end by the breech-piece, and at the other by the projectile. 



Thus, with ;'o and r.^ as the internal and external radii, and 

 jZ>o as the powder-pressure, the longitudinal tension will have 

 its average value 



■^>'o^Poh{^2 



0=A/('-.7V-i) . 



(26) 



tons per square inch. 



The average circumferential tension being 



this longitudinal tension will be 



of the average circumferential tension, reducing to one-half in a 

 thin cylinder, in which we may put r.2 = r^. 



For this reason it was formerly considered safe to leave the 

 longitudinal strength to take care of itself ; but some alarming 

 failures, in which the gun on firing drew out like a telescope, 

 have shown the necessity of carefully hooking the coils together, 

 to provide the requisite longitudinal strength. 



The larger the gun, the greater the number of separate parts 

 requisite in its construction, and the greater the difficulty of 

 providing for longitudinal strength. 



(26) By taking a simple cylindrical tube under given internal 

 and external pressures, and supposing it closed by hemispherical 

 ends, a certain theory of distribution of longitudinal tension can 

 be constructed. 



For while the cylindrical part has the same transverse stresses 

 as previously investigated, the stresses in the hemispherical ends 

 may be considered the same as would be produced if they were 

 joined up into a complete spherical vessel, under the same 

 applied pressures. 



A similar procedure to that already given for the cylinder is 

 shown by Rankine ("Applied Mechanics," § 275) to lead to 

 radial pressure / = ar~^ + b, and tension t = i^ar'^ - b, in all 

 directions perpendicular to the radius r. 



For equation (2) for the cylinder becomes modified in the 

 sphere to 



I 2irrtdr = itri-pi - irr'-p ; (27) 



