Circular Cylinders under certain Practical Systems of Load. 357 



Unwin's (' Testing of Materials of Construction,' p. 419), which is 

 corroborated by Professor Ewing, to the effect that lead, which is a 

 plastic material and flows easily, not only does not hinder expansion 

 of the ends of the block, but forces it. 



It is shown in the paper that, under such conditions, whenever the 

 forced expansion exceeds the natural lateral expansion of the stone 

 or cement, which it practically always does, then the points of failure, 

 instead of being at the perimeter of the ends, are at the centre, and 

 the limiting stress, under these circumstances, may be much less than 

 that obtained for non-expanding ends. Further, this limiting stress 

 depends upon the amount of flow of the lead and has no fixed value — 

 a conclusion confirmed by the experimental results of Unwin. The 

 mill-board test, on the other hand, should give consistent results, 

 although it really introduces too large a factor of safety. The change 

 in the form of the fracture, noticed by Unwin, is also accounted for by 

 theory. 



The values of the apparent Young's modulus and of the apparent 

 Poisson's ratio are investigated. Young's modulus is shown to vary 

 between its true value, when the cylinder is long, and the value of 

 the ratio of stress to axial contraction, when lateral expansion is pre- 

 vented by a suitable pressure, this last corresponding to the case when 

 the cylinder is made very short. 



In the given example, Poisson's ratio is apparently 0*269, the actual 

 value assumed being 0*25. It should diminish down to zero as the 

 cylinder becomes indefinitely short. 



The third problem corresponds to the case of a cylinder whose ends 

 are surrounded by a collar so that the applied torsion couple is 

 transmitted to the inner core by means of transverse shear. 



A general solution is first found for a given arbitrary system of 

 transverse shear. Approximate expressions are given when the length 

 of the cylinder is large compared with its diameter. These show 

 that, to the first approximation, the cross-sections remain undistorted, 

 radii originally straight remaining so. The shear across the section, at 

 any point of it, is connected with the total torsion moment at that 

 section by the same relation as in the ordinary theory of torsion, A 



transverse shear r<j> varying as the square of the distance from the 

 axis exists over the lengths of the cylinder subjected to external 

 stress. 



As a numerical example a cylinder is considered, whose length is 

 7r/2 times its diameter, and which is subjected, over lengths at the 

 ends, each equal to one-fourth of the whole length, to a uniform 

 transverse shear. Using the exact expressions found, the stresses and 

 transverse displacement are calculated for various points, and these 

 are compared with the values calculated from the approximate expres- 

 sions when the cylinder is long. 



