1902.] the Bending of a Beam of Rectangular Cross-section. 495 



load and resting upon a smooth rigid plane, is next considered. The 

 distribution of the pressure upon the plane is investigated and a new 

 form of expansion found for it. It is shown that, outside a certain 

 limited area below the load, a tension is required to keep the elastic 

 solid in contact with the plane, so that such a solid would be lifted at 

 the sides, by applying pressure at the centre. 



(2.) When the stresses across y = ± b are still normal, but are 

 asymmetrical with regard to x = 0. 



In particular the behaviour of a beam under two concentrated loads 

 acting in opposite seDses upon opposite faces of the beam, their lines 

 of action being on opposite sides of the mid-section, is studied. The 

 manner in which the shear across the middle section varies as these 

 loads are made to approach each other is exhibited by various 

 diagrams. These show how rapidly the effects of the particular dis- 

 tribution of any total terminal load die out as we go away from the 

 end. At a distance of the same order as 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 in the paper may give some infor- 

 mation which shall be useful in this connection. 



(3.) When the stresses across y — ± b are purely tangential. The 

 special case here treated is that where these stresses reduce to a 

 single concentrated tangential force. 



As in practice we cannot approximate to a line distribution of 

 shearing stress, the effect of spreading it out over an area is investi- 

 gated. It is then found that, though the displacements are every- 

 where finite and continuous, a discontinuity (though not an infinity) 

 in the surface shear leads to an infinite stress at the point, and is 

 therefore a source of danger to the material. 



It is found also that shear depresses those parts of the solid towards 

 which it acts. Both these results agree with those previously obtained 

 by the author for circular cylinders.* 



The effects of applying tension to a bar by shearing stresses over 

 its faces are considered in this connection. The correction to the 

 readings of an extensometer (which measures the surface stretch), 

 owing to the difference of this distribution of terminal stress from 

 the one usually assumed, is investigated. It is found that no error will 

 be introduced provided no measurements are taken within a distance 

 from the grips less than one and a-half times the long diameter of the 

 section. 



Finally the possible cases of solutions in finite terms are discussed, 

 and such a solution is obtained for a beam which carries a uniform 

 load. It is shown that the assumptions of the usual theory of flexure 

 * < Phil. Trans.,' A, vol. 198, pp. 147—233. 



