EQUILIBRIUM OF AN ISOTROPIC ELASTIC ROD OF CIRCULAR SECTION. 961 



Take S any free mode and S' its conjugate, and suppose for the moment that the 

 coefficients in these are so chosen that their work-difference is unity. 



Let rrii, m/, W2, mj be coefficients such that m^S + ?7^/S' occurs along with 2i in 

 2<Zo, and m.^ + m^'^' along with Sg in z>Zq. 



The work- difference 2itz = z^ of 2^ -I- ?7iiS -f m/S' and S' is equal to the work- 

 difference aX z = Zq of Sg -f TOgS -I- 9772'S' and S' ; 

 i.e. (the work-difference of 2^ and S')-hWi = (the work-difference of Sg and 



which gives mi — m^ ; similarly we find m^ — m^. 



Thus we find the differences of the coefficients with which any particular mode 

 occurs in the various regions in which separate particular solutions are known ; and 

 obviously it is only these differences with which we are concerned in questions of 

 continuity. Clearly, if the body extends to infinity we can make the transitory modes 

 vanish there, and the permanent modes have opposite signs at 2= ± 00. 



The work-differences of all pairs of conjugate modes are determined in Arts. 22 

 and 27. 



A method using Betti's Theorem, similar to that by which 27 (10) is found, is 

 sometimes sufficient to determine the work- difference of a given particular solution and 

 any free mode, when the forces are known ; this is so, for instance, when the forces 

 between ?i and z^ are proportional to an exponential function of z, as in the problem of 

 Art. 3, and the displacements of the particular solution are proportional to the same 

 exponential function. 



45. Relations between the displacements and stresses of any transitory mode. 



We have already (Arts. 22 and 44) made use of the fact that the work-difference at 

 any section of any permanent mode and any transitory mode vanishes. The theorem 

 leads to twelve relations between the displacements and stresses of any transitory mode, 

 which will be here set down in detail. The twelve relations may be arranged in four 

 groups corresponding to the four chief types of deformation of the cylinder [cf. Art. 26). 



Let u^, Uy, u,, zx, zy, zz be the displacements and z-stresses of a transitory mode at 

 any section ; U^, Uj^, U^, ZX, ZY, ZZ those of a permanent mode. Thus 



j j(Ujz + U/y + \J,zz - ZXm, - ZYu^ - ZZu,)dS = 0. 



Here it is evident that the coefficients of the various powers of z must vanish 

 separately. Taking the permanent modes as in Arts. 21, 22, we find : — 



' IJzxdS^O, (1) /"/'.rSr/S = 0, (2) 



JS = 0, (3) 



dS = 0, . (4) 



I I \ —a-{y^ - f) - -^rci^ I zx + dxyzy + E.rw, 



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