EQUILIBRIUM OF AN ISOTROPIC ELASTIC ROD OF CIRCULAR SECTION. 959 



43. Finite cylinder. Any deformation due to end tractions alone is expressible 

 in terms of the free modes for an infinite cylinder. Exact determination of the 

 permanent modes when the end tractions are given. Some problems soluble completely.* 



Once the solution for a force at a single point within the infinite cylinder is known, 

 Betti's Theorem leads immediately to an expression for the displacement at that point 

 in the direction of the force in any deformation whatever, in terms of the body force, 

 the cylindrical surface tractions, and the end tractions and displacements belonging to 

 that deformation. We have only to take the single force solution as first system and 

 the deformation in question as second system in the Theorem, stated in Art. 20. 



The volume integrals and cylindrical surface integrals give a particular solution of the 

 problem for the body force and the cylindrical surface traction. This is the same solution 

 as we have already derived from direct integration of the source solutions. {Cf Art. 24.) 



The surface integrals arising from the ends of the cylinder (which we shall suppose 

 to be z-planes) are, in the three cases of a unit X, Y, and Z internal force, the surface 

 integrals of functions proportional to the three displacements u^, Uy, and u^ in a 

 permanent or transitory free mode of the infinite cylinder. {Cf Art. 23 at the end, 

 and Art. 29 (I), (2)', (3)'.) 



This shows that any deformation of the finite cylinder due to end tractions alone is 

 expressible in terms of the permanent and transitory modes belonging to the infinite 

 cylinder. 



■ An important problem is that in which the end tractions are given and the 

 permanent modes of the solution are required. In this case the modes whose definition 

 in terms of integrals over the ends involves end displacements are simply rigid body 

 displacements, and, omitting these, we have the permanent terms given explicitly in 

 terms of the integral stresses (or stress-7-esultants and stress-couples, cf. Art. 21) at 

 the two ends. That the six chief permanent modes are determinate exactly from the 

 integral stresses at the ends is otherwise evident, for in any transitory mode these must 

 vanish, since they are independent of the section and contain an exponential function 

 of 2; as a factor. 



Certain problems involving given conditions at the ends can be solved completely 

 in a simple form with the help of the solutions for force at a single internal point. 



{a) We can find the symmetrical torsional component of the solution when 

 either displacements or tractions are given at the ends. 



(b) We can find the complete solution when the normal end tractions and 



tangential end displacements are given ; and similarly when 



(c) the tangential end tractions and normal end displacements are given. 



One method of solving these problems is to begin by balancing separately each of 

 the free modes in the source solutions for the given end conditions. {Cf Art. 15, after 

 (3).) For the conditions {a), {b), or (c) it can easily be seen that each /3-mode is 

 balanced by modes associated with the same /3, the coefficients of which may easily be 



* Of. Art. 47. 

 TRANS. ROY. SOC. EDIN., VOL. XLIX. PART IV. (NO. 17). 131 



