CHAPTER IV 



FLEXURE OF BEAMS 



66. Elastic curve. If a beam is subjected to transverse loading, its 

 axis is bent into a curve called the elastic curve. The diff erential equa- 

 tion of the elastic curve is 

 found as follows. 



Let ABDE (Fig. 54) rep- 

 resent a portion of a bent 

 beam limited by two adja- 

 cent cross sections AB and 

 DE, and let C be a point 

 in the intersection of these 

 two cross sections. Then 

 C is the center of curva- 

 ture of the elastic curve 

 FH. Let d/3 denote tlu 

 angle ACE, and through 

 H draw LK parallel to AC\ 

 then the angle LHE is also equal to d/3. Since the normal stress is 

 zero at the neutral axis, the fiber FH is unchanged in length by the 

 strain. Therefore, from Fig. 54, the change in length of a fiber at a 

 distance y from the elastic curve is yd/3 t where dfi is expressed in 

 circular measure. Consequently, the deformation of such a fiber is 



X 



By Hooke's law, = E where p = - ; hence 



Is 



