354 Mr. L. N". G. Filon. On the Elastic Equilibrium of 



conditions in the third. The second case required the introduction of 

 other typical solutions, and the analysis was more intricate. 



The three problems investigated are as follows : — 



In the first I consider a cylinder under pull, the pull not being 

 applied by a uniform distribution of tension across the plane ends, but 

 by a given distribution of axial shear over two zones or rings, towards 

 the ends of the cylinder. 



The second is that of a short cylinder compressed longitudinally 

 between two rough rigid planes, in such a manner that the ends are 

 not allowed to expand. 



The third case is that of the torsion of a bar in which the stress is 

 applied, not by cross-radial shears over the flat ends, as the ordinary 

 theory of torsion assumes, but by transverse shears over two zones or 

 rings of the curved surface. 



The first problem corresponds to conditions which frequently occur 

 in tensile tests, namely, when the piece is gripped by means of pro- 

 jecting collars, the pull being in this case transmitted from the collar 

 to the body of the cylinder by a system of axial shears. 



Analytical solutions are found when this system of axial shears is 

 arbitrarily given, there being given also an arbitrary system of radial 

 pressures. Approximate expressions are deduced when the length of 

 the cylinder is large, compared with its diameter. These show that 

 the strains and stresses may be calculated on the assumption that we 

 have, over any cross-section, a uniform tension across the section, a 

 constant radial pressure and an axial shear proportional to the distance 

 from the axis, the last two occurring only over the lengths of the 

 cylinder where such stresses are applied. The effects of local pressure 

 and shear are thus, for a long cylinder, restricted to a small region 

 and, in the free parts of the bar, we have, to this approximation, the 

 state of things assumed by the ordinary theory. 



In order, however, to study the effect of such a system of surface 

 stresses, when no approximations are involved, I have worked out 

 numerically a case where there is no radial pressure applied externally, 

 and a uniform axial shear is applied between two zones. The solution 

 gives zero tension across the plane ends ; it is not, however, found 

 possible to fulfil completely the condition of no stress, and we have 

 over these limiting planes a self-equilibrating system of radial shears, 

 which, however, will produce little effect at a distance from the ends. 

 The length of the cylinder is taken to be tt/2 times the diameter, this 

 ratio being found to simplify the arithmetic. The two rings of shear 

 extend each over one-sixth of the length and are at equal distances 

 from the mid-section and the two ends. 



In this and the other numerical examples, Poisson's ratio has been 

 taken as one-fourth. This is not correct for most materials, but as the 

 object was to find out the differences between the results of the simple 



