

BENDING MOMENT 293 



where y is the height of the centroid of the area above the neutral 

 ;i\i> and A is the whole area. 



But since a plane section always remains plane, the total force 

 acting on that section must be zero, or the resultant pull must 

 be equal to the resultant thrust. 



E 

 Hence =yA?/ = 



or y = 



Therefore the neutral axis must pass through the centroid of the 

 section. 



Referring again to the elementary strip, 



Total force on the strip = E ~ z 8t/ 



E 



Moment of this force about the neutral axis = ^ y 2 SA 



L\ 



The moment of resistance for the whole section = ^ w 2 8A 



IMM 



E 



= - I 

 R 



where I is the moment of inertia of the section about the neutral 

 axis. 

 Now the moment of resistance = Bending moment, 



E 



Hence M = I 



M E 



IIP = 



I R 



Also, since R is the radius of curvature and the slope of the 

 beam is very small, 1 fi>y 



R" d* 5 



Then M - El S( 



dx z 



In general, for any section of a loaded beam, if 

 I = moment of inertia of the section about the neutral axis 



M = bending moment at the section 



E = Young' s Modulus for the material 



R = radius of curvature at the section 



p = stress induced in the strained fibre 



y = distance of that strained fibre from the neutral axis 



