332 



NATURE 



[July 31, 1890 



tion of the strain, and of uniform lateral tension, {k - %n)e = \e, 

 in every direction perpendicular to the strain. 



(33) These equations, and the previous equations, which show 

 that, when the stress is a simple uniform longitudinal tension P, 

 the strains consist of a uniform extension P/M in the direction 

 of the tension, and of uniform lateral compression ctP/M per- 

 pendicular to the tension, are so fundamental in the theory of 

 the elasticity of isotropic bodies, that we are almost tempted to 

 make a digression here on their proof, in the manner given in 

 Thomson and Tait's "Natural Philosophy," §§ 682, 683, and 692. 



It is necessary to describe and compare the notations carefully, 

 for subsequent purposes, as the variety of notation in the subject 

 of elasticity is very confusing. 



(34) Applying these principles to the gun, we take the three 

 principal directions of stress and strain, as (i. ) circumferentially 

 to the gun, (ii.) radially, (iii.) longitudinally; and now, esti- 

 mating tensions and extensions as positive, we have — 



P = t : 



Q= -P 



ar — 



while the value of R is still indeterminate. 



For the determination of the strains, we denote by tc the 

 increase of radius, r, of a circumferential fibre ; and then i-nti 

 being the elongation of the fibre of original length 2irr, the 

 circumferential extension 



e — 27re</2irr = «/;- ; 



while the radial extension/= dujdr ; the longitudinal extension 

 ^ being as yet undetermined. 



(35) Expressing the strains e and/in terms of the longitudinal 

 tension R, 



M.? = Mu/r =F - (r(Q + R) 



= ar-"' — b + a(ar~- -f- i?') - trR 



= (I + a)ar~- - (l - <j)b - o-R ; 

 or 



M« =: (I -(- a)ar-^ - (i - o)f)r - o-Rr ; . . . (38) 



so that, differentiating with respect to r, 

 Mf^Mduldr 



= - (i + <T)ar-'^ - (I - (r)b - ad{Rj-)/dr 

 = Q - o-P - ad{Rr)/dr. 

 But with 



M/= Q - <r(R -h P), 



Barlow, Lame, and Hart's expressions for the stresses are verified, 

 provided that 



d{Rr)ldr = R ; 

 or 



dRjdr = 0, R = constant. 



(36) On the other hand, expressing the strains e and/ in terms 

 of the longitudinal strain or extension g^ since 



putting 



R = <r(P -f Q) + M^, 

 M^ = yiujr = P - (t{Q -f R) 

 = (I - ff2)P - <7(i + (r)Q - 

 = (I - <r2)(P - <^'Q) - <^M^, 



so that 



Mm = (I - (r'2){(i -{- a')ar-^-(i - ^')br) - <rM.gi- ; . (38*) 

 and differentiating with respect to r, 

 M/= Mduldr 



= (I - (r2)(Q - <t'P) - c:s\d{gr)ldr, 

 agreeing again in giving 



M/= Q - <r(R -(- P, 



provided that 



d{^r)ldr = g; 

 or 



dgldr — o, g = constant. 



NO. 1083, VOL. 42] 



We have proved, then, that either the longitudinal tension R 

 or the longitudinal extension g of the gun must be uniform, for 

 the values of the stresses given by the formulas of Barlow, Lame, 

 Hunt, and Rankine, to be strictly accurate ; we shall follow the 

 ordinary practice in assuming that R is uniform, but the work 

 will be almost precisely the same if we assume that g is uniform 

 (Prof. P. G. Tait, "On the Accurate Measurement of High 

 Pressures," Proc. R.S. Edinburgh, 1879-80). 



(37) Now, let us determine, for the simplest case of the tube 

 A and the jacket B, the requisite shrinkage for producing a given 

 initial pressure SB^ = pi at their common surface ; the shrinkage, 

 denoted by S, being defined as the excess of the outside diameter 

 2p(, of the tube A over the inside diameter 2rj of the jacket B, 

 when both are finished cold in the lathe ; so that 



iS = p. - n. 



The jacket B is now expanded by heat till its inside diameter 

 is greater than 2po, and then slipped over the tube A ; on cool- 

 ing, the jacket B shrinks and grips the tube A with the requisite 

 pressure, /; = ffi^. 



Taking the practical rule that the expansion of steel is one- 

 ten-thousandth for every 15° F., the jacket must be raised in 

 temperature something over 150,000 S/2rj degrees Fahr. 



Denoting by u and v the outward displacement of any circum- 

 ferential fibre of the jacket or tube, of radius r in the jacket and 

 p in the tube ; then, since the tube and jacket fit closely at their 

 common surface, 



p„ -\- Vo = ri + Ui, 

 or 



tii - vo = Po - n = ^S. 



(38) Supposing the tube and jacket to be both of steel of the 

 same quality, so that M, the modulus of elasticity, is the same 

 for both ; and assuming that R is uniform, then in the jacket B, 

 from (38), 



Mwi = {ti + cTpi)ri - o-Rr,-, 



and in the tube A, 



Mi;, = ( - T., 



^e)Po — cRpo 



and now, since pi = Sio, and we may put n = pa, subtraction 

 gives— 



M{t(i - Vo) = (ti + ToVi, 



or 



S = (A + T„)2riim ; (39) 



and ti and t^ having been determined either from the formulas 

 (6) to (16), or graphically from Fig. 3, from the given value of 

 pi — Zo, the requisite value is determined of the shrinkage S, or 

 of S/2ri = (ti + T<,)/M, which is the shrinkage, estimated as a 

 fraction of the diameter. 



This formula shows us that the shrinkage S is the elongation 

 or contraction which would be produced in a bar of steel, one 

 square inch in section, and equal in length to the diameter 2;-;, 

 by a pull or thrust of ti + t,, tons. 



If we had taken g as uniform, we should find in a similar 

 manner — 



S = (i - a-'^)iti + T,)2ri/M (40) 



With steel, a- — \ about, so that or' = ^ ; and the values of 

 the shrinkage calculated on the two assumptions of uniform R 

 and uniform g, would be in the ratio of i to i - cr'^, or as 16 to 

 15 ; thus differing by about 6 per cent., a difference which is 

 practically insensible. 



(39) In the numerical example we have given of the initial 

 stresses of the tube and jacket, ^j = 5, t<, = 5 ; so that 8/2^; = 

 lo/M. 



For gun steel M = 12,600 about (Unwin, "Testing of 

 Materials of Construction," p. 249) ; and supposing the tube and 

 jacket to represent a 3-inch field gun, 2n = 6 ; and then 



S = 1/2 10 = o"oo476, 



476 thousands of an inch. 



(40) In heavy guns, one or more hoops are shrunk on over 

 the jacket; for instance, in the no-ton gun, three such series 

 are superposed. Diagrams in section of modern guns will be 

 found in recent numbers of the Engineer and of Engineering. 



