BEAM OF RECTANGULAR CROSS-SECTION UNDER ANY SYSTEM OF LOAD. 155 
of pressure or of shear, has been discussed. In the case of a beam doubly supported 
and carrying a concentrated load in the middle, a convergent series lias been 
obtained, giving the exact correction which the finite height of the beam makes it 
necessary to apply to Boussinesq’s results for an infinite elastic solid. 
The results of this part of the paper have been tested by experiments on glass 
beams, of which it is hoped to eventually publish an account, and they have been 
found to agree, on the whole, with observation. 
The effects of pressing a block of elastic material which rests on a rigid plane, and 
the manner in which such pressure is transmitted to the plane have also been 
investigated. It has been found that the pressure on the plane is limited to a 
restricted area, outside which the elastic block ceases to be in contact with the plane. 
The effects of shearing stress have next been considered, in particular the 
distortion which it produces in lines parallel to the axis of the beam. As in the case 
of the circular cylinder and in that of the infinite solid bounded by a plane, shear is 
found to depress those parts of the material towards which it acts. 
It is also found that a discontinuity in the shear applied to the surface—although 
the shear remains finite—involves one of the other stresses becoming infinite, and so 
is a source of weakness and danger. 
The behaviour of a beam under two concentrated loads, acting in opposite senses 
upon opposite faces of the beam, has been studied. The manner in which the shear 
across the section varies as these loads are made to approach each other has been 
exhibited by various diagrams. They show how rapidly the effects of the particular 
distribution of any total terminal load die out as we go away from the end. At 
a distance of the order of the height of the beam, they already begin to be 
negligible. 
At a lesser distance than this, however, such effects may become exceedingly 
important. The case of rivets is instanced, and it is suggested that the results 
obtained here may give some information which shall be useful in this connection. 
Finally a solution in finite terms is obtained for a beam which carries a uniform 
load. It is shown that the assumptions of the usual theory of flexure are in this 
case no longer true, but are approximately true only if the height be very small 
compared with the span. The correction to the curvature, as calculated from the 
usual formula, is found to be a constant. 
AVith regald to the numerical work, the arithmetic has been checked wherever 
possible, and it is believed that no serious error has crept in. The values of the 
integrals, however, have been obtained by the use of quadrature formulae, and these 
may not have given a satisfactory approximation in all cases. The three first decimal 
places, nevertheless, should be correct. As the numerical work was undertaken 
chiefly to illustrate fairly large variations and to represent them by diagrams, this 
accuracy appears sufficient. 
X 2 
i*oA3SNTED 
6i£U903 
