“not paral 
the smaller the difference R-7v. The moment ‘of inertia of the whole 
39) 11 | 6x 11)18x 143] 54x . 1859 
MOMENTS AND OCENTROIDS 55 
lel the error in the above value for its moment of inertia is less 
solid is the sum of the moments of inertia of all the rings into which it 
is divided. 
Fig. 60 shows how to convert the section of Fig. 59 into an equivalent 
section symmetrical about an axis perpendicular to XX. 
ab = AB, and cf=CD + EF. 
70. General Method of finding Moments of Inertia of Irregular 
Plane Figures.—T aking a standard rail section (Fig. 61) as an example,* 
let it be required to find the moment of inertia I of the section about an 
axis XX passing through its centre of gravity and perpendicular to its 
one axis of symmetry YY. Divide the section into a number of parallel 
strips, preferably of equal width, perpendicular to YY, and draw the 
centre lines, shown dotted, of these strips. In Fig. 61, 15 strips, each 
#-inch wide, have been taken. Take an axis X,X, perpendicular to YY 
and touching the lower end of the section. Let 2 be the length of any 
oné strip measured at the centre of its width, and let y be the distance of 
its centre line from X,X,. To avoid fractions in this example the linear 
unit is taken in the first instance as one-sixteenth of an inch. The area 
of any one strip is its width multiplied by x, and is denoted by a. Let 
7 denote the distance of the centre of gravity of the section from X,X,. 
Then 7Da= ay, and the moment of 
inertia about X,X,=I,= ay’. The 
work of finding a, MMM ta ee St 
should be tabulated as shown below. SPO BY BE a 
>| 
t 
1 
1 
, 
R 
\ 
| 
{ 
y|z}| a ay ay 
87 | 38 | 6x 38] 18x 1102/ 54x 31958 
81 | 42 | 6x 42] 18x 1134] 54x 30618 
75 | 42 | 6x 42/| 18x 1050| 54x 26250 
69 | 41 | 6x 41] 18x 943)54x 21689 
63 | 22 | 6x 22/)18x 462|54x 9702 
57} 11} 6x 11) 18x 209|54x 3971 
51) 11 | 6x 11) 18x 187) 54x 3179 
45 | 11| 6x 11) 18x 165)54x 2475 
4 
j 
j 
33 | 11 | 6x 11] 18x 121)54x 1331 
27) 11 | 6x 11|18x 99} 54x 891 
21) 11) 6x 11) 18x 77)54x § 539 
15 | 23 | 6x 23) 18x 115|54x 575 
9 | 41) 6x 41) 18x 123)54x 369 
3 | 39 | 6x 39)18x 39) 54x 39 
! 
Totals. } 6 x 365 | 18 x 5969 | 54x 135445 qf Sa ae be 
ee me eee we 
' 
! 
! 
l 
= >a =Zay = Zay? 
J= 
Lay 18x5969 
= as : 365 7 49 sixteenths of an inch. 
49 5 pF : 
= 16 = 3y'5 inches. 1, = eae = 111°6 in inch units, 
= 
* Fig. 61 is half full:size. 
