July 31, 1890] 



NATURE 



133 



The addition of each hoop that is shrunk on modifies the 

 initial stresses previously existing. The annexed diagram (Fig. 7), 

 taken from the American "Notes on the Construction of Ord- 

 nance," Nos. 31, 33, 35, by Lieutenant Rogers Birnie, shows the 

 shrinkage (enlarged 50 times) of the different parts, and the 

 intermediate and final arrangements when a jacket, b, an inner 

 hoop, C, and an outer hoop, D, are successively shrunk on the 

 tube A of the American 8-inch gun, shown in longitudinal 

 section in Fig. 8, 



But knowing the initial stresses in the gun, as determined in 

 the manner already explained in Part I., we can determine the 

 requisite shrinkage at each common surface, for any number of 

 layers, by a formula as simple as that just found for the tube A 

 and jacket B, if only we assume that M, the modulus of elasticity, 

 is the same throughout. 



(41) Denote, as before, by /,„, the radial pressure at the 

 radius, r„„ of the common surface of the mXh. and w» -h ith 

 hoops, as reckoned from the interior ; and by /',«, /», the circum- 

 ferential tensions at the exterior radius, r,„, of the wzth hoo p, and 

 at the interior radius, r„„ of the in -f- ith hoop. 



Denote also by u'm, Um, the outward radial displacement from 

 the unstrained position of the outer surface of the mth hoop, and 

 of the inner surface of the m + ith hoop. 



Then, with uniform R, from (38), 



so that 



M(«„ 



Mm« = (i„, + <Tp,„)r,n - ffRr,„, 



M«'m = {fm + aPmVm - C'R^'m ; 

 ■ u'm) = {fm - t'm)rm. 



STAOES OF 



PARTS PREPARED FOR AStCMBLAQE 



Now, using the notation ^Sm 4 

 unstrained between the mih. and 



to denote the shrinkage when 

 + Ith hoops. 



„Sm -t-i = 2(«„ - u'm) 



= {U - t'm)2rmlU, 





(41) 



the same as for a single tube, A, and jacket, B ; and showing that 

 the shrinkage, mSm + 1, is the elongation or contraction which 

 would be produced in a bar of steel, of modulus of elasticity M, 

 one square inch in section, and of length equal to the diameter 



NO. 1083, VOL. 42] 



I 2r,n, by a pull or thrust of /„ - t'm tons. On the assumption of 

 uniform g, we should find — 



«.S™.H = (I - <r2)(/„, - /'«)2r„./M, . . . (42) 



practically the same as for tiniform R. 



(42) The stress formulas in the mih hoop give — 



2a,n = {pm + t'm)r;n = (pm^ + fm:-l)r^-l. 

 26,n = (pm - t',n = pm-i " fm-l', 



so that 



A,. -/'«. = {U -/«-,)- {pm - Pm-i). 



