160 



KESISTANCE OF MATERIALS 



that this does not make the moduli equal. Assume also that cross 

 sections which were plane before flexure remain plane after flexure 

 (Bernoulli's assumption), which leads to the relation 



where e c and e t denote the distances of the extreme fibers from the 

 neutral axis (Fig. 108). 



Now let the ratio of the two moduli be denoted by n ; that is, let 



E 



Then 



Pt 





FIG. 108 



For a section of unit width the resultant 

 compressive stress R c on the section is 

 R c = \p c e c , and similarly the resultant 

 tensile stress R t is R t =^p t e t . Also, since 

 R c and R t form a couple, R c = R t . Hence 



t) 



p e c = p t e t , or = and, equating this 

 P 



V 

 to the value of the ratio _ obtained above, we have 



Since the total depth of the beam h is h = e c + e t , we have, therefore, 

 e c Ji e c V/i, whence 



h 



S) 



and similarly e t = h *= whence 



vn 



e.= 



h 



1+4. 



Vrc 



Now, by equating the external moment M to the moment of the 

 stress couple, we have 



or = 



