56 APPLIED MECHANICS 
6 x 365 
Area of section = A= ‘ ar = 8°55 square inches. 
492 
I=], -A7=111°6 — 8°55 x 162 = 314 in inch units. 
31°4 
(Radius of gyration)? = 4? = x56 = 3°67, and k= ,/3°67 = 1-92 inches. 
71. Moments of Inertia of Plane Figures made up of Rectangles. 
—A large number of beam and column sections are made up of rectangles 
whose sides are either parallel or perpendicular to the axis about which 
the moment of inertia is required. In such cases the procedure in 
finding the position of the centre of gravity and moment of inertia is as 
follows. Referring to Fig 62, let the rectangle of 
breadth 6 and depth d be one of the rectangles of which +} 
the section is made up, and let X,X, be an axis parallel @ 
to the side b. Let y be the distance of the centre of '! 
the rectangle from X,X,. The area of this rectangle is dd, 
and the area of the whole section is 2jd. The moment 
of the area of this rectangle about X,X, is bdy, and x—_“_, 
> 
y= where 7 is the distance of the centre of Fia. 62. 
gravity of the whole section from X,X,. The moment of inertia of this 
. bb fA 
rectangle about X,X, is Tq + bdy? = bd ( ty), and the moment of 
sas 
inertia of the whole section about X,X, is 2bd (is +7" ) 1. 
The moment of inertia of the whole section about an axis passing 
through its centre of gravity and parallel to X,X, is I, — 7?2bd. 
The particulars for each rectangle and the results of the calculations 
should be tabulated in a form such as is shown below. 
d? 
i2 
@ 
b | d bd y bdy y? oa( i +9" ) 
Totals . .| dd Lhdy zna( 5+) 
72. Transformation of Moments of Inertia—Principal Axes of 
Inertia.—Let EF (Fig. 63) be a plane 
figure, and OA and OB two axes at 
right angles to one another in the 
plane of EF. Let A and B denote 
the moments of inertia of EF about 
the axes OA and OB respectively. 
Let OP and OQ be two other axes in 
the plane of EF, perpendicular to one 
another, and inclined at angles 6 and 
90°+6 respectively to OA. Let P 
and Q denote the moments of inertia of 
