EQUILIBRIUM OF AN ISOTROPIC ELASTIC ROD OF CIRCULAR SECTION. 899 



xT), xz, yz are the components of stress, these being given in terms of the displacements 



Uy;, Uy, u^ tj thc cquRtions 



01/ \ az vx / 



" \ A , T 9^- ^ /9w„ 8mA 



(2) 



where 



. du^ du,, du. 



A = — ^ ^ ? -) ? 



3a; 9// 8z 



In cylindrical coordinates p, w, z the displacements are u^, u^, u^ ; the strains are 



Spp — 



dUp 



_ 1 du^ Up 



p 0(1) p 



SO that 



p oa> dz 82 dp 



. 32<o 1 dWu, ?ifl 3 m, 

 A = — °-\ "! + -i* + ^ 



3p p 3(1) p 3^ 



3m„ ?*„ 1 3?<p 

 dp p p d(D 



(3) 



The stresses are therefore 



— \ * , <i f^Wp " /3?«„ zi„ 1 3w.\ ^ /dup du' 



PP = AA + 2;.^, P-K3p-7^7S^)' ^^ = K^^37 



^. = AA + 2;.^'\ £5 = ^(1 ^^i + Sf"), 



3z \ p 3(0 3^ / 



o.c. = AA + 2p(l^ + ^^) 



p Oil) p ' J 



(4) 



2. General solution of the equations for the displacements, in terms of three 

 independent harmonic functions. 



The equations of equilibrium under no body force are, in terms of displacements, 

 and in rectangular coordinates, 



^3A 3A 3A\ n ly. 



(2) 

 (3) 

 (4) 



The following solutions of these equations are easily verified : — 



/V „ _,8V =_ 9^^ „9^^ ■2(A + 2ja)3^ 



3z? ' "^ ''322 • '"' "^^-'^-^ ^ 



d6 d6 dO 



\ 3a; ' "^3^ 32 ' 



Mj. = — , ?<„ = - — , "2 = ; 



"^ 3y ' •' dx' ' ' 



dxdz dj/dz k + fx dz 



where <^, 0, and \|^ are harmonic functions, that is to say, solutions of Laplace's 

 equation. 



(5) 



We shall use the symbol v for -\ ^-^ = 4(1 - a). 



A + p. 



In (2) A = (2 - vf^ , in (3) and (4) A = . 



32^ 



(6) 



