14:! 
I\rR. L. N. G. FILON OX THE ELASTIC EQUILIBRIUM OF 
Pages 
§ 26. Solution invoh'ing discontinuities at the perimeter of the plane en'Is.217-219 
§ 27. Summary of results.219-221 
§ 28. The thinl prohlem. Case of torsion. Expressions for the displacement and stresses . . . 221-225 
§ 29. Special case of two discontinuous rings of shear. 225-226 
§ 30. Approximations on the lioundary when the cylinder is short. 226-227 
§ 31. Xumerical examjjle. Values of the coefficients and of the displacement and stresses . . 227-230 
§ 32. Discussion of the results. 231 
§ 33. General conclusion. 231-233 
§ 1. Object and Aims of the Paper. 
'Phe nsual solution for the extension and compression of elastic bars assumes that the 
latter are strained under a normal tension or pressure nnifc)rmly distributed across 
the plane ends. In like manner the solution for torsion of such bars assumes that 
the external forces which cause the torsion consist of a determinate system of 
tangential stresses, acting across the plane ends. 
In l)oth cases the solution is such that the torsion and extension are trau.smitted 
througliout the Ijar uuthout change of ti/pe. Such terminal conditions of stress, 
however, do not usually occur in practice, and it accordingly becomes of considerable 
interest to liiul out how the results obtained for such a theoretical system of loading 
are modified, if at all, when we consider applied external stresses which give a closer 
la'presentation of every-day mechanical conditions. 
d’he })resent paper is an attempt towards the solution of this problem in three 
cases, which appear of especial practical interest. 
The first case is that of a bar which is subjected to a determinate system of normal 
]‘adial pressures and of axial shears all over the curved surface, the radial pressures 
being symmetrical about the mid-section and the shears having their sign changed. 
'I'lius the cylinder is subjected to a total axial pull, due to the shears, and also to a 
givei^ transverse ])ressure. The plane ends are free from stress, except for a self- 
equilihrating system of radial shears, which will have little or no effect at points at 
some distance from the ends. 
A special case worked out is that where the normal pressure is zero tliroughout, 
hut a determinate axial shear, which has been taken constant, is made to act over 
two ecpial rings on the surfi\ce of the cvlinder. 
This will give us valuable information about a system of stress which often occurs 
in practice, in testing machines, fiw example, in which a specimen is pulled apart by 
means of pressures applied to the inner rims of projecting collars (see fig. l). The 
shaded })aits of the figur-e rei)resent tlie “grips,” and if S he tlie total pull a])plied, 
tins is transmitted to the test piece by means (fi‘ pressure applied along CA, C'A. 
Now consider the tlnnner cvlinder in the middle ideally produced inside the thicker 
