68 



STRENGTH OF MATERIALS 



due to pure bending strain can be readily calculated by means of 

 the core section, as follows. 



Suppose the external bending moment M lies in a plane perpen- 

 dicular to the plane of the cross section and intersecting it in the 

 line MM. Then, assuming that M is due to an infinitesimal force 

 whose point of application is at an infinite distance from O in the 

 direction OM, the antipolar of this point will be the diameter of the 

 inertia ellipse conjugate to MM. It is proved in analytical geometry 

 that the tangent at the end of a diameter of a conic is parallel to the 

 conjugate diameter. Therefore, if BT is tangent to the inertia ellipse 

 at B, and NN is drawn through parallel to BT, NN will be tin- 

 diameter conjugate to MM. Since tin- 

 greatest stress occurs on the fiber n; 

 distant from the neutral axis, the m;i 

 mum stress will occur at P or R. Thn>uu r h 

 P draw PA parallel to NN and inters- 

 ing MM in A. Then, from Article 59, 



OA OK=OB\ 



or, taking the projections of O.I. "A. and 

 OB on a line perpendicular to AY. 



e- OK >ina 



R 

 M\ 



/N 



FIG. 52 



where e is the perpendicular distance of PA from O. But OB 

 is the distance of the tangent BT from O, and, by Article 49, this 

 distance is the radius of gyration t corresponding t. tin- axis .v.v. 

 Therefore 



(34) e-OKsma = t* = ^> 



where F is the area of the section and I n is its moment of i,, 

 with respect to NN. The component of the external moment M per- 

 pendicular to NN is M sin a. Hence, equating this to the internal 



moment, 



(35) M since =2-2 \\ 



<J' 



where Po is the stress at the distance e from the neutral axis. Sub- 

 stituting in equation (34) the value of / obtained from equation 



