EQUILIBRIUM OF AN ISOTROPIC ELASTIC ROD OF CIRCULAR SECTION. 933 



+ 



u^ = 2<Txy, 



Uy = {z-zf + (r{y'^-x% 

 3= -2(z-z>, 



M^= -{z-z')y, 

 Uy = {z- z')x, 





Mj, = 2(Txy, 



u^= -2{z-z')x, 



x'Z- 



rEa* 



(^'^-^'^<-..y 



+ («,= 1, 2*, = 0, M, = 0){ix'(^'2 + y'2)-(,r + A-T)aV}z(--I 



EaV 



+ (z., = 0, u„ = l, «. = O) I l2/'(.:'2 + y'2) - (cr + 1- - T)aV | z( - -IJ 



+ U, = 0, M, = 0, w,= l ^-'X+^/'Y - 



27rEa2, 



+ I M, = (z - z'), u, = 0, M, = - a; I I (:«'2 - ?/'2)X + 2xV'Y | -^^ . 



For z<j,', change the signs of the above expressions for the displacements throughout. 



The permanent terms for a single force m cylindrical coordinates. 



Similarly in cylindrical coordinates, for a force (P, O, Z) at [p , «', z') we may first 

 take z' = and w' = ; the results are most easily found from (2), (2'), and (6) of Art. 22 

 by changing X, Y, x' , y' into P, O, p\ respectively; finally we change z into z-z' 

 and ft) into oo — w. 



If we denote by S/, S2', . . ., S12' what Si, S2, . . ., S12, Arts. 21 and 22 (1), become 

 when in their forms in cylindrical coordinates we write z — z' for 2 and &) — &)' for ft), then 

 the solution will contain, for z^z\ 



pSi' + 1282' + zs; - Zp'Sg' + op'Sg' 

 + 0^8/+ . . . +012812', 



where, by 22 (6), 



ay= -work done by the force (P, O, Z) acting through the displacements of Si at {p, 0, 0)" 



■ ={>,.-(..|-.>v}z(-J,.). 



1 



(1) 



\ ■ (2) 



"«=~'^'''^2;T^2' '^10 = '^p''" 



Ea*' '^" = '^^"^. 



■Ea* ^^ 



It may be verified easily that these results agree with those stated in Art. 19. 



26. Grouping of the permanent m,odes. 



The twelve permanent modes may conveniently be grouped in four sets, 

 (i) Flexure in the plane zx. S;^, Sg, S7, Sj^. 



«, = F+la-(a;2-.y2) 



f?2F 



u„-^xy---^, 



C?2F 



"'-=~^f ''{r^'^''-^')"("' + D^'4l 



'dz^ 



