RESISTANCE OF MATERIALS 63 



45. Flexure or Bending. When a prismatic rod is fixed at one 

 end and subjected to a stress 

 at the other, it is bent. 

 The top surface will be elon- 

 gated (convex) the lower |* 



shortened (concave) and in 3] -...; 



the intervening space there 3 **--.. 



will be a plane the length of ""., 



which is not altered, called the * % x 



"invariable fibre" or neutral ~- 



axis. The deflection y is, all " 



things being equal, propor- 

 tional to the stress F (fig. 76). 



It can be proved that y varies directly as the cube of the 

 length L of the rod, and inversely as the area a of the section ( J ) 



as well as of the cube of the depth e (fig. 77). Then : 



F. 77. 



If the section is circular, of radius r, the formula will become 



4FL 3 



It will be seen how useful it is to have parts of a large section, 

 and substances of a large modulus of elasticity. It is important 

 to note that the deformation does not extend to the centre. 

 There is, therefore, economy of material in using hollow rods. 

 A hollow rod or tube has a far greater resistance to bending than 

 a solid rod having the same amount of material in its cross section. 



Nature exhibits this economy of material in the tubular 

 structure of the stalks of vegetables, the quills of birds, and in 

 many bones. 2 



( x ) The area a is perpendicular to the direction of the stress ; the depth 

 e is parallel to this direction. 



( 2 ) The observation is that of Galileo (Discorsi e. Dimostraeioni ; sec 

 Opere, vols. viii. and ix. Milan edition, 1811). 



