964 DR JOHN DOUGALL ON AN ANALYTICAL THEORY OF THE 



47. Finite cylinder. Approximate solutions for various conditions at the plane 

 ends* 



The complete solution of any problem for a finite cylinder may be conveniently 

 regarded as composed of three parts : — 



(i) A particular solution for the body forces and cylindrical surface tractions. 

 Such a particular solution is given exactly by the solution for an infinite cylinder which 

 has been defined in the preceding pages. Approximate values of the displacements 

 and of certain of the stresses are set down in Art. 38. Explanations as to the order of 

 magnitude of the error in displacements, strains, and stresses made by taking the 

 formulae of Art. 38 as exact are given in Art. 41. 



(ii) A combination of free permanent modes of the forms set down in Art. 26. 



(iii) A combination of free transitory modes, the influence of which is confined to 

 the immediate neighbourhood of the ends. 



We go on to show how the free permanent modes of (ii) may be determined, exactly 

 or approximately, for various sets of end conditions. 



We will consider the cases when we are given at each end one or other of the sets 



(a) B, zy, 7z, 



(b) M,,. ,u^,£:, 



(c) u, , zx., fi), 



(d) u, , u„ , u, , 



not necessarily the same set at both ends. The conditions which occur in most of the 

 ordinary practical problems can be brought under these heads. For (a), (6), and (c) 

 the exact relations given at the end of Art. 45 enable us to determine the permanent 

 modes exactly. 



(-) (I) and (II) give g^ and g, (V) and (VI) ^ and ^, (IX) ^, and 



(XI) -J-, at an end where zx, zy, and zz are known. 



(h) (II) and (IV) give F and ^, (VI) and (VIII) G and ^, (IX) ^, and 



ciz az^ dz 



(XII) n, at an end where u^, Uy, and zz are known. 



(c) (I) and (III) give ^ and |^, (V) and (VII) ^ and ^, (X) H, and (XI) 



-y-, at an end where u,, zx, and zy are known. 

 We have, moreover, at any point between the ends 



|^(F,G) = 0. and J(H.n) = 0. 



The functions F, G, H, and TI are not always completely determinate from these 

 conditions, but any indeterminateness is easily seen to amount to no more than a rigid 

 body displacement. 



* Of. Art. 43. 



