ANALYSIS OF STRESS IX BEAMS 



67 



is essential that the point of application of the load shall lie within 

 the core section. 



Consider a rectangular cross section of breadth b and height h. 

 For the gravity axes MM and NN (Fig. 51) the corresponding mo- 

 ments of inertia are 



m 12 



Hence the radii of gyration are 

 b 



Vl2 



.2887J 



A 



and 



and 





= == = .2887 h, 

 Vl2 



I 



and the inertia ellipse is constructed on 

 these as semi-axes. To determine the core 

 section it is sufficient to find the antipole 

 of each side of the cross section PQRS. 

 Suppose A is the antipole of PQ, B the 

 antipole of PS, etc. Then, by Article 60, 

 the antipole of any line through P, such 

 as LL, lies somewhere on Alt ; that is to 

 say, as the line PQ revolves around P to 

 th* position PS, its antipole moves along 

 . I // from A to /?. The core section in the 

 present case is thus found to be the rhom- 

 bus A B CD. 



Fi-Min Article 59, OC- OK = OT* = 4"' since tne semi-axes of the 



\'l i j 



ellipse are the radii of gyration. But OK = - ; hence OC = - and 



7 7 



AC = -- Similarly, BD = - This proves the correctness of the 

 3 3 



rule ordinarily followed in masonry construction, namely, that in order 

 to insure that the stress shall all be of the same sign, the center of 

 pressure must fall within tin 1 middle third. of the cross section. 



63. Calculation of pure bending strain by means of the core 

 section. Let Fig. 52 represent the cross section of a beam subjected 

 to pure bending strain. In this case the neutral axis passes through 

 the center of gravity <>f a cross section, and therefore, from Article 60, 

 the strain can be considered as due to an infinitesimal force at an 

 in tii lite distance from the origin. Under this assumption the stress 



