ANALYSIS OF STKESS IX BEAMS 



45 



4.9 in. 2 . Consequently, the breadth of the lower flange of the equivalent homo- 

 geneous section is 



16.9 + 4.9 



.76 



= 29.1 in. 



The distance of the center of gravity of this equivalent section below the top 

 is found to be 7.69 in., and its moment of inertia about the gravity axis OZ is 

 in.* (Fig. 30). 



49. Inertia ellipse. Dividing equation (20) by F and expressing 

 the result in terms of the radii of gyration by means of equation (23), 



(24) 



= cos 2 a + t\ sin 2 a, 



win-re t and t n are the radii of gyration with respect to the axes of 

 Y and Z respectively, and t a is the radius of gyration with respect to 

 a gravity axis inclined at an angle a to the axis of Z. 



Now let / be a length defined by the relation 



I. Then 



t v = - , t t = - ; and substituting these values of t v and t t in equa- 



tion <24), it becomes 



or, dividing by fi, 



A I I a 9 ' '.I o 



r m = cos 3 a + -, siirT, 



This is the equation of an ellipse 

 with semi-axes /, and t ut called the 

 inertia ellipse, the coonli nates of 

 any point of the curve being / cos a 

 and / sin a. 



By means of the inertia ellipse 

 tin- moment of inertia with re- 

 spect to any gravity axis AB (Fig. 

 31) can be obtained as follows. ^ 



The equation of a tangent to the ellipse + ^ = 1 at the point 



tt Cv 



Fio. 81 



/ 





(25. 



ell? + yy'a* - a'P = 0. 



