46 



STRENGTH OF MATERIALS 



It is proved in analytical geometry that in order to reduce the linear 

 equation Az + By + C = to the normal form z cos ff+ y sin ft - c = 

 it. is necessary to divide throughout by V^ a + B\ Applying this 

 theorem to equation (25), it becomes 



z'b* , y'a* aW_ ^ = 0j 



Vz'^ 4 + yv* " v^^Tyv Vz'v + yv 



where 



= cos /S, 



= sin A 



/^ 6 4 + y v v^v+yv 



Substituting these values in the expression a 2 cos 2 + 6* sin 1 A it 



becomes 



y'V 



' + y v 



whence, since ft = a > 



c 2 = a 2 cos 2 -f & 2 sin 2 /9 = a 2 sin 2 a -f 6* cos^a. 

 Since the semi-axes of the inertia ellipse are a = t t and 

 expression becomes __/ /i 



or, comparing this expression with equation (24), 



The radius of gyration corresponding to any gravity axis AB can 

 therefore be found by drawing a tangent t tin- 

 inertia ellipse parallel to ^4/* t and measurinL: tin- 

 distance of this tangent from the cent- 



Since the inertia ellipse is constructed on tin* 

 j// principal radii of gyration as semi-axes, it can In- 

 drawn on all the ordinary forms of cross sect inn. 

 and when this is done the method given al 

 greatly simplifies the calculation of the m<>m<-nt 

 of inertia with respect to any gravity axi- which 

 FIG. 32 is not a principal axis. 



Problem 50. From the Carnegie handbook of structural steel tin- principal 

 radii of. gyration of T-shape, No. 72, size 3 in. by 4 in., are l.ii.'J in. and ..".'. in. 

 Construct the inertia ellipse (Fig. 32). 



