1870.] Mr. W. H. Barlow on the Resistance of Beams, 



345 



for me by Mr. W. Barrett Davis, by the aid of a grant from the Donation 

 Fund at the disposal of the Royal Society. The paragraphs of the present 

 memoir are numbered consecutively with those of the former memoirs on 

 Qualities. 



II. " On the Cause and Theoretic Value of the Resistance of Flexure 

 in Beams." By W. H. Barlow, F.R.S. Received April 13, 

 1870. 



(Abstract.) 



The author refers to his previous papers, read in 1855 and 1857, wherein 

 he described experiments showing the existence of an element of strength 

 in beams, which varied with the degree of flexure, and acts in addition 

 to the resistance of tension and compression of the longitudinal fibres. 

 It was pointed out that the ratio of the actual strength of solid rectan- 

 gular beams to the strength as computed by the theory of Liebnitz is, 

 In cast iron, as about 2\ to 1 . 

 In wrought iron as If and If to 1. 

 And in steel, as If and If to 1. 

 The theory of Liebnitz assumes a beam to be composed of longitudinal 

 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 homogenous elastic solid, must not be understood as 

 meaning filaments of the material. In effect it represents lines of direc- 

 tion, in which the action of forces can be ascertained and measured ; for 

 in torsion-shearing and " angular deformation " 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 

 v 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 compres- 

 sion and extension of the diagonal fibres constitute an element of strength 

 equal to that of the longitudinal fibres, and that flexure is the conse- 

 quence of the relative extensions and compressions in the direct and dia- 

 gonal fibres, arising out of the amount, position, and direction of applied 

 forces. 



Pursuing the subject, it is shown that certain normal relations subsist 



