156 STRENGTH OF MATERIALS 



and the unknown moment will be denoted by M . On any other sec- 

 tion AB the bending moment M and single force P' are respectively 



p 



M = Mo + - (p - p cos ), 



(102) p 



P' = - cos A 



in which p is the radius of the mean fiber and j3 is the angle which 

 the plane of the section AB makes with the base CD. 



Now, no matter whether the section is flattened or elongated by 

 the strain, from the symmetry of the figure the diametral sections 

 MN and PP will always remain at right angles to one another. 

 Therefore the total angular deformation A/8 for the quadrant under 

 consideration must be zero ; that is to say, 



' A/3 = 0. 



'o 

 But, from Article 66, 







Consequently, 



Inserting in this expression the value of M obtained above, 



or 



whence 



M * = 2-7T Pp = 

 which is the maximum negative moment. 



From formula (102), the maximum positive moment must occur 

 when cos ft = 0, that is, when /S = ^ , or at top and bottom. Therefore 



-^max = -M; = = .318 P/X 



2 ^ 



