to Transverse Strain in Beams. 399 



resistances being in the direction of the fibres, they are 

 measured with, and undistinguishable from, the resistance to 

 the elongation of the several filaments. 



But if these form part of the extended portion of a bar 

 subjected to transverse strain, and therefore undergoing 

 flexure, as in fig. 4, the extension of the middle fibre being 

 assumed to be the same as that of all the fibres in fig. 3, the 

 distance between b and c, b and c', &c. will be increased to a 

 greater extent than in the previous case of direct tension, and 

 therefore an increased resistance developed between them. 

 But, on the other hand, the distance between a and b, a' and b, 

 &c. will be less increased, and the resistances thereby caused 

 will be less. So that unless the bending or flexure be ex- 

 tremely great, the aggregate resistances to the displacement 

 of the particles of the middle fibre B will be the same, neither 

 more nor less than that to its elongation to an equal amount 

 by direct tension. 



These considerations appear to present some a priori ground 

 against the supposition of such a resistance to flexure as is 

 assumed by Mr. Barlow, and they are further confirmed by 

 the investigations of M. St. Venant*, referred to and endorsed 

 by Sir William Thomson in his article on Elasticity in the 

 Encyclopaedia Britannica. 



M. St. Venant proved that, with the important practical 

 exception of a thin flat spring, the resistances to the flexure 

 of a rod or bar consist in " the mutual normal forces pulling 

 the portions of the solid towards one another in the stretched 

 part and pressing them from one another in the condensed 

 part, and that the amount of this negative or positive normal 

 pressure per unit of area must be equal to the Young's 

 modulus (E) at the place multiplied into the ratio of its dis- 

 tance from the neutral line of the cross section to the radius 



of curvature Hence the principal flexural rigidities are 



simply equal to the product of the Young's modulus 



into the principal moment of inertia of the cross section" f; 

 that is to say, the moment of the resistances is the sum of the 

 moments of the tensile and compressive stresses. 



This had indeed been assumed, as Sir William Thomson 

 remarks, without proof by earlier writers ; but it was 

 St. Venant who first gave the subject " satisfactory mathe- 

 matical investigation," and " proved that the old supposition 

 is substantiallv correct." 



* Memoires des Savants Etrangers, 185/5, " De la Torsion des Prismes, 

 avec considerations sur leur Flexure." 



t Encyc. Brit. vol. vii., art. " Klastieitv." 



2 E 2 



