490 X. STATICS OF DEFORMABLE BODIES. 



= I C(gX,-xZ,}dS= -Ec<i C C 



126) M= (gX,-xZ,}dS= -Ec<i x*dxdy = - Eaj y 



where I y is the moment of inertia of the cross -section about the 

 ]T-axis. 



The line of centers of mass x = y = becomes strained into 



ft P^ 



x' = - -y-> a parabola, or, since the displacement is supposed small, 

 a circle of radius This displacement is called uniform flexure, for 



the curvature of the central line is constant. It is produced by the 

 action of no forces, but of a couple applied at the ends. The couple 

 is the same for all cross -sections, and is equal to the product of 

 Young's modulus by I y and the curvature of the central line 



127) M 



This is the theorem of the bending moment. Such a strain is produced 

 in a bar when we take it in our hands and bend it by turning them 

 outwards. If the bar has a length I from the fixed section, the 

 deflection of the end is 



128) M = _ 



and the flexural rigidity, or moment per unit displacement per unit 

 of length is 2EI y . 



For a rectangular section of breadth 6 and height h, 



I= 



=J J 



_A _A 

 2 



For a circular section of radius R, 



For a circular and rectangular beam of equal cross -sections, since 

 = bh the ratio of stiffnesses is 



^rectangle _ bh s 



and if & = 7& 



Circle ' 12 



^circle 



^ = 1.0472. 



Since w = a^xz, a plane cross -section at a distance from the 

 origin remains a plane which is rotated through the angle = a^z, 



