3o6 



NA TURE 



[July 24, 1890 



ever thick, cannot stand, if unsupported, an internal pressure 

 greater than the working tenacity of the material. 



But, as the pressures in gunnery often exceed the tenacity of 

 any known material, the requisite strength must be provided by 

 an initial compression of the tube due to shrinking on one or 

 more cylindrical jackets, 



(9) F'g- 2 is drawn representing graphically the state of stress 

 set up in a tube A by an external applied pressure as,,, as in the 

 tube or flue of a boiler by the external pressure of the water, or 

 in the intertal tube of a gun by the shrinkage pressure of the 

 outside jacket. 



Denote by pi and po the inner and outer radii of the tube A, 

 and by p any intermediate radius. 



The stress at any point of the tube will now consist of a radial 

 pressure £8, and of a circumferential pressure r, represented by 

 the ordinates of the curves as^asas,, t^ttj ; and dividing the tube 

 by a diametral plane popiOpipo, and considerins; the equilibrium 

 of inch length of either laalf, we shall find as before that the area 

 fi-iTtroPo = the rectangle 03!^ — as„po ; while considering the 

 equilibrium of any coaxial cylindrical portion, bounded by the 

 radii po and p, then the area pTT„pt, = rectangle OaSo - rectangle 

 Oa) ; or, in the notation of the Integral Calculus — 



rdp =■ Siopo — Sip ; 



(") 



leading, by differentiation with respect to p, to 



T = - d{sip)/dp; (II*) 



the general solution of which can, as before, be exhibited in the 

 form — 



ffi = j3 + op-2, T = ;3 - o/)-*, . . . . (12) 

 or 



(as - T)p2 = 2a, as + T = 2fi, 

 where o and $ are arbitrary constants, determined from the values 



of the arbitrary pressures applied to the interior and exterior 

 surfaces. 



(10) Now with (Si = o. 



so that 

 and then 



o = y3 + api 



ap — ^ = 

 ap-^ = 



-■' - Po 



Pi 



Pt"" - P< 



2pi - 2 





Po - Pi' 

 20„^ 



Pi 



Po- 



■ .(13) 



. .(IS) 

 . .(16) 



Given, then, t; the maximum allowable crushing pressure 

 of the material, then 



Apo^ - pr)l2pj^ = Ir^i - piVpo') 



(17) 



is the maximum allowable external pressure on the tube. 



(11) If we makep,, = n and a(„ = /;, the tube a of Fig. 2 

 may be supposed to be gripped by the cylinder B of Fig. i, of 

 which only the upper halves need now be shown, as in Fig. 3 ; 

 and now Fig. 3 will represent the cross-section of a tube. A, 

 over which a jacket, B, has been shrunk, as at the breech end 

 of an ordinary field-gun, and will represent graphically the 

 stresses set up when the pressure, as^ = /», at the common sur- 

 face, is supposed known ; these are called the initial stresses, or 

 stresses of repose ; the internal pressure at the radius p,- and the 



external pressure at the radius ro being zero, as the atmospheric 

 pressure is insensible in our calculations. 



In Fig. 3 we notice that the total pull resistance across the 

 section r^rj, represented by the area rototiVi, is equal to the total 

 thrust resistance of the section popu represented by the area 

 .PoToTipi, and each of these is equal to the resultant pressure 

 thrust represented by the area of the rectangle O/,-. 



(12) Now, suppose a pressure P (say 15 tons on the square 

 inch) is applied at the interior of the tube, either by the steady 

 pressure of -water, as in a hydraulic press, or by the momentary 

 pressure of gunpowder, as in the bore of a gun. 



We suppose that the additional stresses due to this pressure, 

 P, which we shall call the poivder stresses, are the same as those 

 which would be set up in a homogeneous cylinder of internal 

 radius pu and external radius Vo, by a steady pressure, P ; and 

 these powder stresses will therefore, by what precedes, in equa- 

 tions (6), (7), (8) (Fig. i), at a distance r from the axis, consist 

 of a radial pressure— 



. p ,._2_,.^-2 ^^^^ 



and a circumferential tension- 



Pv 2 - To 



(>9) 



NO. 1082, VOL. 42] 



having a maximum value at the bore of 



We must superpose these powder stresses on the initial 

 stresses of the compound cylinder to obtain the stresses whei 

 the cylinder is used as a gun (or hydraulic press) ; these are 

 called the firing stresses, and they are exhibited graphically in 



FiS- 4- . . . . , 



(13) We now see the reason for settmg up mitial stresses m 



the gun by shrinking a jacket over the interior tube. 



For the maximum circumferential tension at the bore on firing 



is reduced by the initial stresses from 



T = pP-i: 



Pi' 



vPi 



(20) 



while at the interior of the jacket the circumferential tension is 

 altered from 



toP': 



^^^^7F^ 



(21) 



The maximum stresses in the gun are thereby equalized to a 

 great extent, and material can be econo.iiized. 



