146 



STRENGTH OF MATERIALS 



for the analysis of stress, as in the theory of straight beams; that is 

 to say, it will be assumed that the stress is proportional to the 

 deformation produced, and that any plane section remains identical 

 with itself during the deformation. 



Since the fibers on the convex side are longer than those on t bi- 

 concave side, it will take less stress to deform them an equal amount. 

 Therefore the neutral axis does not pass through the center of gravity 

 G of the section, but through some other point 7), below G, as shown 

 in Fig. 102. For if the neutral axis passed through G, the total 



deformation above and h-lo\v 

 fi would be of equal amount, 

 and therefore the total stress 

 above G would be less than 

 that below G t siii< the libers 

 above G are longer than those 

 below. This shifting of tb,- 

 neutral axis constitute the 

 fundamental difference be- 

 tween the t IK-MI y ,,f straight 

 and curved pieces. 



Now li-t tb,. length of anv 

 fiber, such as .17 A in I i- 10*2, 



be denoted by /, and the distance of this fiber from a gravity udfl 02 

 by y. Also, let p denote the radius of curvature OG of the piece, & 

 the angle between two plane sections, and a the angle of deformation 

 of a plane section. Then 



/ = ft- MO =(00 + GN)/3=(p + y)ff, 

 and the deformation dl of the fiber MN is 



dl = NN 1 = a-NI>=(y + d)a, 



where d denotes the distance GD between the neutral axis and the 

 gravity axis. From Hooke's law, 



FIG. 102 



whence 



I 



Let - = k, where & is a constant. Then this expression for p reduces to 



