Mr. W. H. Barlow on the Resistance of Flexure in Beams. 131 



The theory of Liebnitz assumes a beam to be composed of longi- 

 tudinal fibres only, contiguous, but unconnected, and exercising no 

 mutual lateral action. But it is remarked that a beam so constituted 

 would possess no power to resist transverse stress, and would only 

 have the properties of a rope. 



Cast iron and steel contain no actual fibre ; and wrought iron 

 (although some qualities are fibrous) is able to resist strain nearly 

 equally in any direction. 



The idea of fibre is convenient as facilitating investigation ; but the 

 word fibre, as applied to a homogeneous elastic solid, must not be 

 understood as meaning filaments of the material. In effect it repre- 

 sents lines of direction, in which the action of forces can be ascer- 

 tained and measured ; for in torsion-shearing and " angular defor- 

 mation " the fibres are treated by former writers as being at the 

 angle of 45°, because it has been shown that the diagonal resistances 

 have their greatest manifestation at that angle. 



Elastic solids being admitted, to possess powers of resistance in the 

 direction of the diagonals, attention is called to omission of the effect 

 of resistance in the theory of beams. 



The author then states, as the result of his investigation, that com- 

 pression and extension of the diagonal fibres constitute an element of 

 strength equal to that of the longitudinal fibres, and that flexure is 

 the consequence of the relative extensions and compressions in the 

 direct and diagonal fibres, arising out of the amount, position, and 

 direction of applied forces. 



Pursuing the subject, it is shown that certain normal relations 

 subsist between the strains of direct fibres and their relative diago- 

 nals, evenly distributed strain being that in which the strain in the 

 direct fibres is accompanied by half the amount of strain in the rela- 

 tive 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 com- 

 pressive, 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 compressive forces acting longitudinally on either side 

 of the neutral plane, and show3 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. 



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