346 



On the Resistance of Flexure in Beams. 



[May 19, 



between the strains of direct fibres and their relati ve diagonals, evenly 

 distributed strain being that in which the strain in the direct fibres is 

 accompanied by half the amount of strain in the relative diagonal fibres. 



Any disturbance of this relation indicates the presence of another force. 



Thus tensile forces applied at right angles to compressive forces of 

 equal amount, produce no strain in the diagonals. But if forces applied 

 at right angles to each other are both tensile, or both compressive, the 

 strain in the diagonal is as great as that in the direct fibres. 



It is also pointed out that in a given fibre a b c, the point b may be 

 moved with regard to a and c, thus producing plus and minus strains in 

 the same fibre. 



Treating a solid as being made up of a series of laminae, and showing 

 that every change of figure can be represented by the variation in length 

 of the diagonals, taken in connexion with those of the direct fibres, the 

 author proceeds to trace the effects of the application of tensile and com- 

 pressive forces acting longitudinally on either side of the neutral plane, 

 and shows that curvature is the result of the relation between the strains 

 in direct fibres and those in the diagonals. 



The operation of a single tensile force applied along one side of the 

 plate and a transverse stress are likewise traced out, and the conditions 

 of " elastic equilibrium" referred to. 



The amount of resistance offered by the diagonal fibres is shown as 

 follows : — 



abed represents a portion of a beam strained by transverse forces 

 into the circular curve a e. 

 Two resistances arise. 



1. That due to the extension and compression of the longitudinal fibres 

 produced by the rotation of b d about the neutral axis, which is the 

 resistance considered in the theory of Leibnitz. 



2. That due to the extension and compression of the diagonal fibres, 

 caused by the deformation of the square abed into the figure ahoc, 

 which is the resistance of flexure. 



It is then shown that in a solid rectangular beam, the second resist- 

 ance is equal to the first, and that both resistances act independently, and 



