126 STRENGTH OF MATERIALS 



bending moment M '. Then, if p denotes the radius of curvature, : 



Article 66, 



Again, suppose that this circular cylinder is made to assume the 

 flattened form as the result of an additional bending moment M t and 

 let p f denote the corresponding radius of curvature. Then 



1 = M'+ J/ 

 p' ~ El 

 Consequently, 



(66) }'~~p = Jll' 



From the differential calculus,* 



If the deformation is small, -- is intiniU'simal,and r differs infinitesi- 



aa 



mally from a. Therefore, neglecting intiniU'simals of an order higher 

 than the second, the expression for p' becomes 



and, consequently, 



- 1 /- 



a - 



Since p = a, - = -, and therefore 

 p a 



(67) 



p' p a 2 da* 



Comparing equations (66) and (67), 



(68) l.^-i^L 



a 2 da* ~ El 1 



* Calculus, p. 163. 



beCaUSe the calculus expression for p' contains a square root in 



