432 
Proceedings of the Royal Society of Edinburgh. [Sess. 
XXVIII. — On the Friction at the Extremities of a Short Bar 
subjected to a Crushing Load, and its Influence upon the 
Apparent Compressive Strength of the Material. By G. H. 
Gulliver, B.Sc., A.M.I.Mech.E., Lecturer in Engineering in the 
University of Edinburgh. 
(MS. received November 30, 1908. Read December 21, 1908.) 
1. Crushing Strength. 
In making crushing tests of various constructional materials the usual form 
of test-piece employed is that of a small cylinder or prism, having a height 
of 1 to 2 diameters. The pressure is applied to the bases of the specimen 
by means of two plates of steel or of cast iron, which are somewhat harder 
— that is, less easily deformable — than the material under test. The friction 
between these plates and the ends of the specimen necessitates, for the 
deformation and rupture of the piece, a load somewhat greater than if the 
friction did not exist. So far as the writer is aware, no investigation of 
the effect of this end friction has been published hitherto, and the following 
is offered therefore as giving an approximate solution to the problem. The 
method employed is simple. In any bar, two portions about to experience 
a relative movement along their common plane of sliding are regarded as 
two rigid bodies in a condition of static equilibrium, and the sum of all the 
forces acting on each piece is accordingly equated to zero. Since the 
external forces are known, or can be determined, the internal resistance to 
sliding, under certain assumptions, may be calculated. The method is 
admittedly imperfect, and the degree of approximation of the calculated 
results to actual conditions is uncertain ; but these results are not un- 
interesting, and they show, if not grossly incorrect, the importance of the 
influence of the end friction upon the apparent crushing strength of a bar. 
The assumptions made with regard to internal friction in a previous 
paper (1) are made here also. The coefficient of friction is supposed to be 
independent of the load ; and a cohesive force, K, is assumed to act normally 
to the surface of each particle of the metal, giving rise to an internal 
frictional resistance to sliding, equal to //K. It is assumed, further, that 
the external frictional resistance to deformation at the loaded ends of the 
specimen can be regarded as equivalent to a force applied to the piece in a 
direction tangential to these surfaces. In all cases the bars are supposed to 
