Circular Cylinders wider certain Practiced Systems of Load. 355 



and the modified theories, rather than to calculate the absolute stresses 

 and displacements for any given material, the exact value of Poisson's 

 ratio adopted was comparatively unimportant. 



It is then found that the stress is greatest at the points where the shear 

 is discontinuous, i.e., at the ends of the collar in a practical case. At 

 these points it is theoretically infinite. This result is true whatever 

 the dimensions of the cylinder. For materials like cast iron or hard 

 steel, which are brittle, such points would therefore be those of greatest 

 danger ; but in such a case as that of wrought iron or mild steel, for 

 instance, the stress will be relieved by plastic flow. 



The tensile stress varies considerably over the cross-section, and the 

 distortion of the latter is large. Towards the middle of the bar, the 

 axial displacement at the surface is, roughly, twice what it is at the 

 centre. 



In tensile experiments the elongation is usually measured by the 

 relative displacement of two points on the outer skin of the cylinder, 

 as recorded by an extensometer. When the test-piece is seized in this 

 way, the surface stretches more than the interior, and consequently a 

 negative correction should be applied to the readings of the extenso- 

 meter. In the somewhat extreme case considered, this correction may 

 amount to as much as 30 per cent, 



The lateral contraction is very much smaller than the theory of 

 uniform tension indicates, being in fact never so great as 60 per cent, 

 of the amount calculated on that hypothesis. For points inside the 

 material the discrepancy is still greater. These variations appear due 

 to the fact that there are considerable radial and cross-radial tensions 

 inside the material, these tensions being often equal to about one-fifth 

 of the mean tension Q, which would give the same total pull. 



Tables are given in the paper showing the values of the radial and 



axial displacements u and w, and of the four stresses rr, zz, rz, 

 (in the notation of Todhunter and Pearson's ' History of Elasticity,' 



st being the stress, parallel to s, across a face perpendicular to t) for 

 points in the cylinder at distances from the axis = 0, "2a, *4a, "6a, a ; 

 a being the radius of the cylinder ; and for intervals of length parallel 

 to the axis equal to tenths of the half-length. These tables are 

 illustrated by curves and diagrams. 



The second problem is of considerable importance, as it illustrates 

 the crushing of blocks of cement or stone, when they are compressed 

 between iron planes, or between sheets of mill-board, so that their ends 

 are constrained not to expand. 



The analytical solution is made up, partly of a finite number of 

 terms which are algebraic and rational in r and z, and partly of infinite 

 series involving sines and cosines containing z. By suitably combining 

 these two types of terms all the conditions can be satisfied. 



The numerical example taken was one in which the length is nearly 



