424 PROBLEMS 



PROBLEM 6. MOMENT OF INERTIA OF AN AREA. 



(a) Problem. Calculate the moment of ineltia, I, of a standard 

 9" [@ 13.25 Ibs., about an axis through its center at right angles to 

 the web: (i) By Culmann's approximate method; (2) by Mohr's 

 approximate method; (3) by the algebraic method. Omit the fillets. 

 Scale of channel, i" == i". Scale of forces, i"= I sq. in. 



(b) Methods. Divide the channel into convenient sections and 

 consider the areas as forces acting through their centers of gravity, (i) 

 Culmann's method (Fig. 22). Start force polygon (a) at (3.5", 9.1") 

 and take pole at (5.45", 4.6"). Draw equilibrium polygon (b). Now 

 with intercepts a-b, b-c, c-d, d-e, e-f, f-g, g-h, h-i, as forces, and a 

 new pole at (4.5", o.i") construct equilibrium polygon (d). The 

 moment of inertia is (approximately) I = H X H' X y- (2) Mohr's 

 method (Fig. 23). Calculate the area of the equilibrium polygon (b) 

 by means of the planimeter or by dividing it into triangles and (ap- 

 proximately) 1 = area equilibrium polygon (b) X 2 H. If the area 

 is divided into an infinite number of sections, or if the true curve of 

 equilibrium be drawn through the points determined, this method gives 

 the true value of 7. (3) Algebraic method. The moment of inertia 

 about the center line is / = /'+ A d * + 2 1" where /' = moment of 

 inertia of the main rectangle ; A = area of the two flanges ; d = distance 

 of the center of gravity of the flanges from the center line ; and I" = 

 moment of inertia of each flange about an axis through its center of 

 gravity parallel to the center line. 



(c) Results. The algebraic method gives the true value of /; 

 Mohr's method gives a value more nearly correct than Culmann's 

 method, as would have been expected. The values of / given in the 

 various hand-books are calculated by the algebraic method. 



PROBLEM 6a. MOMENT OF INERTIA OF AN AREA. 

 (a) Problem. Calculate the moment of inertia, I, of a standard 

 9" [@ 1 S Ibs., about an axis through its center at right angles to 

 the web: (i) By Culmann's approximate method; (2) by Mohr's 

 approximate method; (3) by the algebraic method. Omit the fillets. 

 Scale of channel, i" = i". Scale of forces, i" = i sq. in. 



